THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


\j 


The  RALPH  D.  REED  LIBRARY 


DKPARTMKNT  OF  GEOLOGY 

UNIVER.S1TY  of  CALIFORNIA 

L08  ANGELES,  CAUF. 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/courseinmathemat01woodiala 


A  COURSE  IN 

MATHEMATICS 


FOR  STUDENTS  OF  ENGINEERING  AND 
APPLIED  SCIENCE 


BY 

FEEDERICK    S.  WOODS 

AND 

FREDERICK  H.  BAILEY 


Professors  of  Mathematics  in  the  Massachusetts 
Institute  of  Technology 


Volume  I 

ALGEBRAIC  EQUATIONS 

FUNCTIONS  OF  ONE  VARIABLE,  ANALYTIC  GEOMETRY 

DIFFERENTIAL  CALCULUS 


GINN  AND  COMPANY 

BOSTON     •    NKW    YORK     •    CHICAGO     •    LONDON 
ATLANTA     •     DAT.T.A8    •    COLUMBU8     •    SAN    KKANCISCO 


Copyright,  1907,  by 
Frederick  S.  Woods  and  Frederick  H.  Bailby 


ALL  RIGHTS  RESERVED 
620.12 


Cbe  gtbenttum   l^rtet 

GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


Geology 
Library 

PKEFACE 


This  book  is  the  first  volume  of  a  course  in  mathematics 
designed  to  present  in  a  consecutive  and  homogeneous  manner 
an  amount  of  material  generally  given  in  distinct  courses  under 
the  various  names  of  algebra,  analytic  geometry,  differential 
and  integral  calculus,  and  differential  equations.  The  entire 
course  covers  the  work  usually  required  of  a  student  in  his  first 
two  years  in  an  engineering  school,  the  first  volume  containing 
the  work  of  the  first  year.  In  arranging  the  material,  however, 
the  traditional  division  of  mathematics  into  distinct  subjects  is 
disregarded,  and  the  principles  of  each  subject  are  introduced  as 
needed  and  the  subjects  developed  together.  The  objects  are  to 
give  the  student  a  better  grasp  of  mathematics  as  a  whole,  and 
6f  the  interdependence  of  its  various  parts,  and  to  accustom  him 
to  use,  in  later  applications,  the  method  best  adapted  to  the 
problem  in  hand.  At  the  same  time  a  decided  advantage  is 
gained  in  the  introduction  of  the  principles  of  analytic  geometry 
and  calculus  earlier  than  is  usual.  In  this  way  these  subjects 
are  studied  longer  than  is  otherwise  possible,  thus  leading  to 
greater  familiarity  with  their  methods  and  greater  freedom  and 
skill  in  their  application. 

In  carrying  out  this  plan  in  detail  the  subject-matter  of  this 
volume  is  arranged  as  follows: 

1.  An  introductory  chapter  on  elimination,  including  the  use 
of  determinants.  This  chapter  may  be  postponed  or  omitted,  if  a 
teacher  prefers,  without  seriously  affecting  the  subsequent  work. 

2.  Graphical  representation.  Here  the  student  learns  the  use 
of  a  system  of  coordinates  and  the  definition  and  plotting  of  a 
function. 

3.  The  study  of  the  algebraic  polynomial.  This  includes  the 
analytic    geometry    of    the    straight    line,    the    more    important 


t  GS' 


iv  PREFACE 

theorems  of  the  theory  of  equations,  and  the  definition  of  a 
derivative.  Simple  applications  of  the  calculus  to  problems 
involving  tangents,  maxima  and  minima,  etc.,  are  given.  In  this 
way  a  student  obtains  an  introduction  to  the  piinciples  of  the 
calculus,  free  from  the  difficulties  of  algebraic  computation. 

4.  The  study  of  the  algebraic  function  in  general.  The  knowl- 
edge of  analytic  geometry  and  calculus  is  here  much  extended 
by  new  applications  of  the  principles  already  learned.  Simple 
applications  of  integration  are  also  introduced.  The  study  of  the 
conies  forms  part  of  the  work  in  this  place,  but  other  curves  are 
also  used  and  care  is  taken  to  avoid  giving  the  impression  that 
analytic  geometry  deals  only  with  conic  sections ;  in  fact,  the 
chapters  which  deal  especially  with  the  conies  may  be  omitted 
without  affecting  the  subsequent  work. 

5.  The  study  of  the  elementary  transcendental  functions.  It 
has  been  thought  best  to  assume  the  knowledge  of  elementary 
trigonometry,  since  that  subject  is  often  presented  for  admission 
to  college,  —  a  tendency  which  should  be  encouraged.  Tlie 
chapter  discusses  the  graphs,  the  differentiation  of  transcendental 
functions,  and  the  solution  of  transcendental  equations. 

6.  The  work  closes  with  chapters  on  the  parametric  represen- 
tation of  curves,  polar  coordinates,  and  curvature.  In  the  first  of 
these  chapters  the  solution  of  locus  problems,  which,  from  some 
standpoints,  is  the  most  important  part  of  analytic  geometry, 
finds  its  natural  place ;  for  this  problem  involves,  in  general, 
the  expression  of  the  coordinates  of  a  point  on  a  locus  in  terms 
of  an  arbitrary  parameter,  and  possibly  the  elimination  of  the 
parameter. 

As  compared  with  the  usual  first  course  in  analytic  geometry, 
there  will  be  found  in  this  volume  fewer  of  the  properties  of  the 
conic  sections,  'except  as  they  appear  in  problems  set  for  the 
student.  On  the  other  hand,  a  greater  variety  of  curves  are 
given,  and  it  is  believed  that  greater  emphasis  is  placed  on  the 
essential  principles.  All  work  in  three  dimensions  is  postponed 
to  the  second  year,  and  is  to  be  taken  up  in  the  second  volume 
in  connection  with  functions  of  two  or  more  variables,  partial 
differentiation,  and  double  and  triple  integration. 


PREFACE  V 

This  volume  contains  the  matter  usually  given  in  a  first  course 
in  differential  calculus,  with  the  exception  of  differentials,  series, 
indeterminate  forms,  partial  differentiation,  envelopes,  and  some 
advanced  applications  to  curves.  These  subjects  will  find  their 
appropriate  place  in  the  further  development  of  the  course  in 
the  second  volume.  Integration  has  been  sparingly  used  as  the 
inverse  operation  of  differentiation,  and  without  employing  the 
integral  sign.  Simple  applications  to  areas  and  velocities  are  given. 
To  do  more  would  require  the  expenditure  of  too  much  time  on 
the  operation  of  integration,  and  the  introduction  of  too  many 
new  ideas  into  one  year's  work.  The  integral,  as  a  limit  of  a 
sum,  with  its  many  applications,  will  form  an  important  part  of 
the  second  year's  work. 

In  the  preparation  of  the  text  the  needs  of  a  student  who 
desires  to  use  mathematics  as  a  tool  in  engineering  and  scientific 
work  have  been  primarily  considered,  but  it  is  believed  that  the 
course  is  also  adapted  to  the  student  who  studies  mathematics 
for  its  own  sake.  Abstract  discussions  are  avoided  and  frequent 
applications  and  illustrations  are  given.  Illustrations,  however, 
which  are  beyond  the  range  of  a  first-year  student's  knowledge 
of  physical  science  are  omitted.  The  proofs  are  made  as  rigorous 
as  the  maturity  of  the  student  will  admit.  It  is  to  be  remembered 
in  this  connection  that  the  earlier  chapters  are  to  be  studied  by 
students  w^ho  have  just  entered  college. 

In  the  preparation  of  the  book  the  authors  have  had  the  advice 
and  criticism  of  the  mathematical  department  of  the  Massachu- 
setts Institute  of  Technology.  In  particular,  they  are  indebted 
to  the  head  of  the  department.  Professor  H.  W.  Tyler,  at  whose 
invitation  the  book  has  been  written,  and  whose  suggestions  have 
been  most  valuable. 

Massachusetts  Institute  of  Technology 
September,  1907 


CONTENTS 


CHAPTER  I  — ELIMINATION 

Article  Page 

1,  2.  Determinant  notation 1 

3.  Properties  of  determinants    . 6 

4.  Solution  of  n  linear  equations  containing  n  unknown  quantities, 

when  the  determinant  of    the  coefficients  of    the   unknown 
quantities  is  not  zero 12 

5.  Systems  of  n  linear  equations  containing  more  than  n  unknown 

quantities 15 

6.  Systems  of  n  linear  equations  containing  n  unknown  quantities, 

when  the  determinant  of   the  coefficients  of  the  unknown 
quantities  is  zero 17 

7.  Systems  of  linear  equations  in  which  the  number  of  the  equa- 

tions is  greater  than  that  of  the  unknown  quantities      ...     18 

8.  Linear  homogeneous  equations 21 

9.  Eliminants 23 

Problems 25 


CHAPTER   II  —  GRAPHICAL  REPRESENTATION 

10.  Real  number 28 

11.  Zero  and  infinity 29 

12.  Complex  numbers 31 

13.  Addition  of  segments  of  a  straight  line 32 

14-15.  Projection 34 

16.  Coordinate  axes 35 

17.  Distance  between  two  points 36 

18-19.  CoUinear  points 38 

20.  Variable  and  function 40 

21.  Classes  of  functions 43 

22.  Functional  notation 44 

Problems 45 

vii 


viii  CONTENTS 

CHAPTER  III  —  THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 
Article  Page 

23.  Graphical  representation 50 

24-26.  The  general  equation  of  the  first  degree 52 

27.  Slope 54 

28.  Angles 55 

29.  Problems  on  straight  lines 58 

30-31.  Intersection  of  straight  lines 61 

32.  Distance  of  a  point  from  a  straight  line 63 

33.  Normal  equation  of  a  straight  line 64 

Problems 65 

CHAPTER  IV — THE  POLYNOMIAL  OF  THE  ^Yth  DEGREE 

34-36.  Graph  of  the  polynomial  of  the  second  degree 70 

37.  Discriminant  of  the  quadratic  equation 73 

38.  Graph  of  the  polynomial  of  the  nth  degree 74 

39.  Solution  of  equations  by  factoring 77 

40-41.  Factors  and  roots 78 

42-43.  Number  of  roots  of  an  equation 80 

44-45.  Conjugate  complex  roots 82 

46.  Graphs  of  products  of  real  linear  and  quadratic  factors  ...  83 

47.  Location  of  roots 86 

48.  Descartes'  rule  of  signs 87 

49-51.  Rational  roots .89 

52.  Irrational  roots 92 

Problems 94 

CHAPTER  V  —  THE  DERIVATIVE  OF  A  POLYNOMIAL 

53.  Limits 97 

54.  Slope  of  a  curve 99 

55.  Increment 100 

56.  Continuity 101 

57.  Derivative .102 

58.  Formulas  of  difFerentiation 103 

59.  Tangent  line 104 

60.  Sign  of  the  derivative 106 

61.  Maxima  and  minima 108 

62.  The  second  derivative 110 

63.  Newton's  method  of  solving  numerical  equations 114 

64.  Multiple  roots  of  an  equation 116 

Problems 118 


CONTENTS  ix 

CHAPTER  VI  —  CERTAIN  ALGEBRAIC  FUNCTIONS  AND 

THEIR  GRAPHS 

Article  Page 

65-66.  Square  roots  of  polynomials 121 

67.  Functions  defined  by  equations  of  the  second  degree  in  ?/    .     .  127 

68.  Functions  involving  fractions      .  • 128 

69.  Special  irrational  functions 131 

Problems 133 

CHAPTER   Vll  —  CERTAIN  CURVES  AND  THEIR  EQUATIONS 

70-72.  The  circle 134 

73-75.  The  ellipse 139 

76-78.  The  hyperbola 142 

79-80.  The  parabola 146 

81.  The  conic 148 

82.  The  witch 149 

83.  The  cissoid 151 

84.  The  strophoid 152 

85.  Examples 154 

Problems 155 

» 

CHAPTER  Vni  —  INTERSECTION  OF  CURVES 

86.  General  principle 161 

87-89. /i(x,y)  =  0  and /2(x,.?/)  =  0 161 

90.  fi(x,y)='0  iiTidf„(x,y)  =  0 10(3 

91. /„(z,  ?/)  =  0and/,(x,  .y)=0 168 

92-93.  If^(x,y)  +  kf„(x,y)=0 171 

Problems 175 

CHAPTER  IX  —  DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

94.  Theorems  on  limits        178 

95.  Theorems  on  derivatives         179 

96.  Formulas 184 

97.  Derivative  of  u" 185 

98.  Higher  derivatives .  187 

99.  Differentiation  of  implicit  algebraic  functions        188 

100.  Tangents 190 

101.  Normals • 191 

102.  Maxima  and  minima 192 

103.  Point  of  inflection 194 


X  COKTENTS. 

Akticle  Page 

104.  Limit  of  ratio  of  arc  to  chord 195 

105.  The  derivatives  —  and  -^        .     ; 196 

(Is  d,i 

106.  Velocity 198 

107.  Components  of  velocity 200 

108.  Acceleration  and  force 202 

109.  Other  illustrations  of  the  derivative 203 

110.  Integration 205 

Problems 209 


CHAPTER  X  —  CHANGE  OF  COORDINATE  AXES 

111.  Introduction 217 

112-114.  Change  of  origin  without  change  of  direction  of  axes      .     .  217 

115.  Change  of  direction  of  axes  without  change  of  origin      .     .  221 

116.  Oblique  coordinates 223 

117.  Change  from  rectangular  to  oblique  axes  without  change  of 

origin 224 

118.  Degree  of  the  transformed  equation 225 

Problems 225 

CHAPTER  XI  —  THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE 

119.  Introduction 229 

120.  Removal  of  the  xy-tevm 229 

121.  The  equation  Ax^  +  By'^  +  2  Gx  +  2  Fy  +  C  =  0      .     .     .     .231 

122.  The  limiting  cases 234 

123.  The  determinant  AB  -  H^ 235 

124.  The  discriminant  of  the  general  equation 236 

125.  Classification  of  curves  of  the  second  degree 237 

126-127.  Center  of  a  conic 238 

128.  Directions  for  handling  numerical  equations 240 

129.  Equation  of  a  conic  through  five  points 241 

130.  Oblique  coordinates 244 

Problems 244 

CHAPTER  XII  — TANGENT,  POLAR,  AND  DIAMETER  FOR  CURVES 
OF  THE  SECOND  DEGREE 

131.  Equation  of  a  tangent 246 

132.  Definition  and  equation  of  a  polar 247 

133.  Fundamental  theorem  on  polars 247 


CONTENTS  xi 

Article  Page 

134.  Chord  of  contact 248 

135.  Construction  of  a  polar 249 

136.  The  harmonic  property  of  polars 249 

137.  Reciprocal  polars 251 

138-140.  Definition  and  equation  of  a  diameter 252 

141.  Diameter  of  a  parabola 254 

142.  Parabola  referred  to  a  diameter  and  a  tangent  as  axes    .     .  255 

143.  Diameters  of  an  ellipse  and  an  hyperbola 256 

144.  Conjugate  diameters 258 

145.  Ellipse  and  hyperbola  referred  to  conjugate  diameters  as  axes  259 
146-147.  Properties  of  conjugate  diameters 260 

Problems 262 

CHAPTER  XIII  — ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

148.  Definition 266 

149.  Graphs  of  trigonometric  functions 266 

150.  Graphs  of  inverse  trigonometric  functions 269 

151.  Limits  of and '- — 270 

h  h 

152.  Differentiation  of  trigonometric  functions 272 

153.  Differentiation  of  inverse  trigonometric  functions  .     .     .     .276 

154.  The  exponential  and  the  logarithmic  functions 279 

155.  The  number  e 280 

156.  Limits  of  (1  +  A)*  and  ^1^ 283 

157-159.  Differentiation  of  exponential  and  logarithmic  functions      .  284 

160.  Hyperbolic  functions 288 

161.  Inverse  hyperbolic  functions 291 

162.  Transcendental  equations 293 

Problems 296 

CHAPTER  XIV  —  PARAMETRIC  REPRESENTATION  OF  CURVES 

163.  Definition 302 

164.  The  straight  line 302 

165.  The  circle 303 

166.  The  ellipse 303 

167.  The  cycloid 305 

168.  The  trochoid 306 

169.  The  epicycloid 307 

170.  The  hypocycloid 309 


xii  CONTENTS 

Article  Page 

171.  Epitrochoid  and  hypotrochoid      , 309 

172.  The  involute  of  the  circle 311 

173.  Time  as  the  arbitrary  parameter 313 

174.  The  derivatives 315 

175-176.  Applications  to  locus  problems 316 

Problems 323 

CHAPTER  XV  — POLAR  COORDINATES 

177.  The  coordinate  system 329 

178.  The  spirals        331 

179.  The  conchoid 334 

180.  The  limacon 336 

181.  The  ovals  of  Cassini 338 

182.  Relation  between  rectangular  and  polar  coordinates    .     .     .  341 

183.  The  straight  line 342 

184.  The  circle 342 

185.  The  conic,  the  focus  being  the  pole 343 

186.  Examples 344 

187.  Direction  of  a  curve 345 

188.  Derivatives  with  respect  to  the  arc 347 

189.  Area 348 

Problems 349 

CHAPTER  XVI  — CURVATURE 

190.  Definition  of  curvature 353 

191-192.  Radius  of  curvature        354 

193.  Coordinates  of  center  of  curvature 356 

194.  Evolute  and  involute 357 

195.  Properties  of  evolute  and  involute .  359 

196.  Radius  of  curvature  in  parametric  representation  ....  360 

197.  Radius  of  curvature  in  polar  coordinates 361 

Problems 362 

Answers 365 

Index 381 


A  COURSE  m  MATHEMATICS 


CHAPTEE  I 

ELIMINATION 

1.  Determinaiit  notation.  Elimination  is  the  process  of  obtain- 
ing from  a  certain  number  of  equations  containing  two  or  more 
unknown  quantities  one  or  more  equations  which  do  not  contain 
all  of  these  quantities.  The  quantities  removed  are  said  to  have 
been  eliminated.  The  solution  of  equations  is  essentially  the  elim- 
ination of  all  but  one  of  the  unknown  quantities.  The  process  of 
elimination  leads  to  the  formation  of  certain  expressions  in  the 
coefficients,  for  which  a  special  name  and  a  corresponding  notation 
have  been  invented.  In  this  chapter  we  shall  consider  equations  of 
the  first  degree,  or  linear  equations.  These  are  equations  in  which 
no  term  contains  more  than  one  unknown  quantity,  and  that  in 
the  first  degree. 

Ex.1.  aix  +  6i2/ +  ci  =  0, 

a2X  +  62^  +  C2  =  0. 

To  eliminate  y,  multiply  the  first  equation  by  62,  the  second  by  —  61,  and 
add.  To  eliminate  x,  multiply  the  first  equation  by  —  a^,  the  second  by  ai, 
and  add.    There  results 

(ai62  -  a26t)  a;  +  (C162  -  C261)  =  0,  /o\ 

(aib2  —  a^bi)  y  +  (aiC2  —  OiCi)  =  0. 

Unless  0162  —  (tibi  =  0,  equations  (2)  give  at  once  the  solution  of  (1). 
If  0162  —  a^bi  =  0,  the  method  used  to  eliminate  y  also  eliminates  x,  and  the 
equations  need  further  discussion,  to  be  given  in  §  6. 

1 


ELIMINATION 


Ex.  2. 


ai«  +  hy  +  ciz  +  di  =  0, 
a^x  +  b^y  +  CiZ  +  d2  =  0, 
Osx  +  bsV  +  C3Z  +  dz  =  0. 


(1) 


To  eliminate  y  and  2,  multiply  the  first  equation  by  {baCs  —  ftgCa),  the  second 
by  -  {bid  -  bsci),  the  third  by  (61C2  -  62C1),  and  add.    There  results 

[ai(&2C8  -  bsCn)  -(h{biC3  -  bgCi)  +  03(6100  -  ftgCi)]* 

+  [di  (62C8  -  bsCi)  -  d^  (61C3  -  63C1)  +  da  (61C2  -  62C1)]  =  0, 

or        (oibjCs  +  ajftsCi  +  OsftiCa  —  aihc^  —  Oa^iCa  —  036201)  x 

+  {dibuPs  +  d^ci  +  ds&iCa  -  ^16300  -  ^06103  -  dzb^Ci)  =  0.  (2) 

To  eliminate  z  and  «,  multiply  the  first  equation  by  —  (0203  —  a^c^),  the 
second  by  (OiCg  —  agCi),  the  third  by  —  (ciCj  —  0201),  and  add.    There  results 

{ai62C8  +  a263f-"i  +  036102  —  016302  —  O261C8  —  036201)  y 

+  (aid2C8  +  a^fiaCi  +  asdiC2  —  axd^c^  —  a^diCs  —  03^201)  =  0.  (3) 

To  eliminate  x  and  y,  multiply  the  first  equation  by  (0263  —  0362),  the  second 
by  —  (O163  —  0361),  the  thii-d  by  (0162  —  O261),  and  add.    There  results 

(O1&2C8  +  O263C1  +  036102  —  O163C2  —  026103  —  O362O1)  z 

+  (aibjds  +  0263(^1  +  0361^2  —  Oi63d2  —  0261^3  —  0362^1)  =  0.  (4) 

Equations  (2),  (3),  and  (4)  give  the  solution  of  (1),  unless 

Oi&aCs  +  OibzCi  +  036102  —  016302  —  O261C3  —  O362C1  =  0. 

The  exceptional  case  will  be  considered  in  §  6. 

The  binomials  which  occur  in  the  solution  of  Ex.  1  are  called 
determinants  of  the  second  order.    The  symbol 


is  used  to  denote  the  determinant  afi^—  afi^.    Then  equations  (2) 
of  Ex.  1  may  be  written 


«i    ^1 


»  + 


=  0, 


«i    ^1 


'y  + 


a,    c, 
2 


1    "1 


=  0. 


DETERMINANT  NOTATION  3 

The  polynomials  which  occur  in  the  solution  of  Ex.  2  are  called 
determinants  of  the  third  order.    The  symbol 


«1 

\ 

Ci 

^2 

h 

^2 

«3 

h 

Cs 

is  used  to  denote  the  determinant 

«AC8+  aA(^l+  «8*1^2—  «^A^2—  «2^lC8—  CCsK^^l- 


The  results  of  Ex.  2  may  then  be  written 


a. 

\ 

Ci 

a^ 

h 

^2 

«  + 

a. 

K 

Cs 

C?j       ftj      Cj 

d^     &2    Cg 

=  0, 

C?3            63            Cg 

«i 

h      Ci 

«2 

h       ^2 

2/  + 

«8 

i^3        Cs 

«! 

d. 

«1 

«2 

d. 

^2 

^8 

d. 

Cz 

=  0, 


«1 

h 

H 

a. 

h 

C2 

2  + 

«8 

h 

^8 

«1 

\ 

d. 

^2 

K 

d. 

«8 

h 

ds 

=  0. 


By  the  work  of  Ex.  2, 


a^    &j    Cj 

^2      ^2      «2 


=  a. 


^2      «2 

^3        ^3 


^  ^1 


-fa. 


5„    c„ 


which  may  be  taken  as  the  definition  of  a  determinant  of  the 
third  order. 

Similarly  a  determinant  of  the  fourth  order  is  indicated  by  the 
symbol 


a,    b,    c, 

d. 

^2       \       «2 

d. 

(^S       K       ^8 

d. 

a,    h,    c. 

d. 

and  is  defined  as  equal  to 


&2    C2    d^ 

h      ^3      ^3 

?>.    c,    d. 


I  ^1    ^1    d^ 
-a\\    C3    <^3 


+  a. 


^ 

c,     d^ 

?>2 

^2        C?2 

-«4 

h 

c,    c?. 

^ 

Cl 

< 

&2 

^^2 

d. 

^'3 

^3 

d. 

4  ELIMINATION 

If  now  each  of  these  determinauts  of  the  third  order  is  expressed 
in  terms  of  determinants  of  the  second  order,  we  shall  have  finally 
the  determinant  of  the  fourth  order  expressed  as  an  algebraic 
polynomial  of  twenty-four  terms. 

2.  In  general  a  determinant  of  the  nth  order  is  an  algebraic 
polynomial  involving  n^  quantities,  called  elements.  The  symbol 
of  the  determinant  is  obtained  by  writmg  the  elements  in  a  square 
of  n  rows  and  n  columns.  If  in  such  a  symbol  a  row  and  a  col- 
umn are  omitted,  there  is  left  the  symbol  of  a  determinant  of  the 
next  lower  order.  This  new  determinant  is  said  to  be  a  minor  of 
the  original  determinant,  and  is  said  to  correspond  to  the  element 
which  stands  at  the  intersection  of  the  omitted  row  and  column. 
We  shall  now  give  as  definition : 

A  determinant  is  equal  to  the  algebraic  sum  of  the  products 
obtained  by  multiplying  each  element  of  the  first  column  by  its 
corresponding  minor,  the  signs  of  the  products  being  alternately 
plv^  and  minus. 

By  repeated  application  of  the  same  definition  to  the  minors 
obtained,  we  eventually  make  the  value  of  the  determinant  depend 
upon  determinants  of  the  second  order,  and  thus  obtain  the  poly- 
nomial indicated  by  the  original  symbol. 

Students  who  desire  a  more  general  definition  and  discussion  of 
determinants  are  referred  to  treatises  on  the  subject.  We  shall 
derive  here,  as  simply  as  possible,  only  those  properties  which  are 
of  use  in  solving  equations.  Before  doing  so,  however,  we  need  to 
show  that  the  word  "column"  may  be  changed  to  "row"  in  the 
above  definition,  thus :  A  determinant  is  also  equal  to  the  sum  of 
the  products  obtained  by  multiplying  each  element  of  the  first  row 
by  the  corresponding  minor,  the  signs  of  the  products  being  alter- 
nately plus  and  minus. 

For  a  determinant  of  the  third  order  the  student  may  verify 
that  '. 


=  a. 


-&. 


-^-Cx 


DETERMIKANT  NOTATION  6 

The  theorem  thus  shown  to  he  true  for  a  determinant  of  the 
third  order  may  be  proved  for  one  of  the  fourth  order  as  follows : 


=  a. 


=  «1 

C2  ^2 

C3    d, 
c,    d^ 

-«2 

h 

c,    d, 
C3    d,^ 
c,    d] 

+  ^3 

\    d, 

2    ^2 

'4       ^4 

-a^ 

b^    c^    d, 
K    c,    d^ 
h    C3    d^ 

(by  definition) 

K    C2    d^ 
K    C3    d^ 
b,    c,    d^ 

-ah    ""^    ^^ 

-Ci 

63      £?3 

^4    d. 

+  d. 

\      «3     1 
^4      ^4    J 

H   '  c,    d. 

-^1 

K    d, 
b,    d. 

+  d^ 

&2      C,    1 
^4      <^j 

-Ci 

^2    ^'  +d 

\      C2I 
^3      ^3    J 

(as  already  proved) 

\    c^    d^ 

h    C3    ^3 
b,    c,    d^ 

H  '  G,  d. 

-«3 

^2     d-2 

c,  d. 

+  a. 

^2     ^2    "1 

C3    d^   f 

r    ^-i  d„ 
.     ^'^V^b\d\ 

-«3 

K  d„ 
\d. 

K  d.  1 

-.{ 

'            ^3 

"C3 
C4 

-^3 

\      «2 

K  C4 

+  ^4 

h    ^2 

^3      ^3 

} 

(by  a  rearrangement) 


=  a. 


h  C2  d^ 

^2  ^2  ^2 

^2     ^2     ^2 

«2     ^2     ^2 

h  C3  ^3 

-h 

^3     C3     ^3 

+  c, 

«3     *3     ^3 

-d. 

<^3     h     h 

64  C4  d. 

^4    C4    ^4 

^4     &4    d^ 

^4     &4     C4 

(by  definition) 

In  a  similar  manner  the  theorem  may  be  proved  successively 
for  determinants  of  the  fifth,  the  sixth,  and,  eventually,  any  order. 


g  ELIMINATION 

3.  Properties  of  determinants. 

1.  A  determinant  is  unchanged  in  value  if  the  rows  and  the 
columns  are  interchanged  in  such  a  manner  that  the  first  row 
becomes  the  first  column,  the  second  row  the  second  column,  and 
so  on. 

The  student  may  verify  that 


«1        «2 


«1 

\ 

«i 

as 

h 

^2 

= 

*8 

K 

^3 

a. 


a„ 


c,     c„ 


This  proves  the  theorem  for  determinants  of  the  second  and 
the  third  orders.  To  prove  it  for  one  of  the  fourth  order,  proceed 
as  follows : 


a^  6j  Cj  d^ 


^2  h  ^2  ^2 
C'S  h  ^i  ^3 
^4    ^4    ^4    ^4 


\  c,  d^ 

*1   Ci   <^l 

6,    C,    d. 

^1  «i  ^1 

=  «! 

h  Cs  ds 

-«2 

^3   ^'S   ^8 

+  ^3 

h    C2    ^2 

-a^ 

^2   ^2   ^2 

K  C4  ^4 

\   C,   rf. 

\  c,  <^, 

h   Cs   ^3 

«!    ^2   «8   «4 

h    b    h   b 

h  h  K 

^    ^'3    ^'4 

^1    ^2    ^4 

^1     ^2    *3 

t'l  ''2  "s  "4 
C,    C^   C3  c, 

d^  d^  d^  d^ 

=  «i 

^2     C3    C4 

-^2 

C,   C3   c. 

+  ^3 

C,     C^    C, 

-a. 

<^1     ^2     ^3 

^2   ^3   < 

^1  <  ^4 

^1   C?2   ^4 

^1   ^2   ^3 

The  expressions  on  the  right  of  these  equations  are  equal,  and 
hence  the  determinants  of  the  fourth  order  are  equal.  In  the  same 
manner  the  theorem  may  be  proved  for  determinants  of  higher 
order. 

It  follows  from  this  theorem  that  any  property  which  is  true  of 
the  rows  is  also  true  of  the  columns,  and  vice  versa.  The  following 
theorems  are  stated  for  both  rows  and  columns,  but  are  proved  for 
the  rows  only. 


PROPERTIES  OF  DETERMINANTS 


2.  If  two  consecutive  rows  {or  columns)  of  a  determinant  are 
interchanged,  the  sign  of  the  determinant  is  changed. 

The  student  may  verify  that 


a^    b„ 

= 

=  — 

a„ 

5 

«1 

\ 

«i 

K       h       ^2 

«i 

h 

^1 

a^ 

^ 

^i 

a^ 

\ 

^'2 

= 

«1       ^        ^1 

> 

«2 

h 

^2 

=  — 

% 

&3 

C3 

^3 

h 

^3 

a 

3    ^3 

^3 

^3 

h 

^^3 

«2 

K 

^2 

The  theorem  is  then  proved  for  determinants  of  the  second  and 
the  third  orders.  To  prove  it  for  a  determinant  of  the  fourth  order, 
consider 


\    c, 

d. 

h       ^2 
^3       ^3 

d, 

d. 

and 

I,    c. 

d. 

h  C3 


By  definition, 


«j  h^  Cj  d 


a,  &2  ^2  do 

%    &3  ^^3  ^3 

«^  l^  c^  d^ 

a^  h^  Cj  c?j 

«,   &3  Cg  6?3 

^2    ^2  ^2  ^2 

«4    ^4  C4  f?^ 


&2    C2    ^2 

^1   ^1    ^1 

^1    ^1    ^1 

\    Cj    d^ 

=  «1 

^3    C3    f^3 

-S 

^3   C3   ^5 

+  «3 

^2    «2   <^2 

—  a^  b^  Cg  d^ 

&4    C4    ^4 

^4   C4   f?4 

I,  c,  d^ 

h  C3  ^3 

K  C3  ^3 

\  ^x  dx 

^1    ^1    ^1 

&^  Cj  c?^ 

=  «i 

^2  ^2  ^2 

-^3 

&2  Cg  d^ 

+  «2 

63    C3    (^3 

-«4 

&3    C3    C?3 

^'4    C4    «^4 

K  H  d\ 

^4    C^    ^4 

^2    ^2   ^2 

Comparing  these  two  expressions,  it  will  be  noticed  that  the 
minors  which  multiply  a^  and  a^  (the  elements  of  the  unchanged 
rows)  differ  in  the  two  expressioils  by  the  interchange  of  two  con- 
secutive rows,  and  that  the  minors  which  multiply  a^  and  a^  (the 
elements  of  the  interchanged  rows)  are  the  same  in  the  two  expres- 
sions but  are  preceded  by  opposite  signs.  It  is  evident  on  reflection 
that  these  laws  always  hold ;  and  hence,  if  the  theorem  is  true 
for  determinants  of  any  order,  it  is  true  for  determinants  of  the 


8 


ELIMINATION 


next  higher  order.    The  theorem  is  known  to  be  true  for  determi- 
nants of  the  third  order ;  hence  it  is  universally  true. 

3.  A  determinant  is  equal  to  the  algebraic  sum  of  the  products 
obtained  by  multiplying  each  element  of  any  row  {or  column)  by  its 
corresponding  minor,  the  sign  of  each  product  being  ]}lus  or  minus 
according  as  the  sum  of  the  number  of  the  row  and  the  number  of 
the  column  in  which  the  element  stands  is  even  or  odd. 

For  the  hth.  row  may  be  made  the  first  row  of  a  new  determi- 
nant by  ^  —  1  interchanges  of  two  consecutive  rows.  By  theorem  2, 
if  k  is  odd,  the  new  determinant  is  equal  to  the  original  one ;  and 
if  k  is  even,  the  new  determinant  is  equal  to  minus  the  original  one. 
The  new  determinant  may  now  be  expressed  by  definition  as  the 
algebraic  sum  of  the  elements  of  its  first  row  multiplied  by  their 
mmors,  which  are  the  same  as  those  of  the  ^th  row  of  the  original 
determinant.  Hence  the  original  determinant  is  equal  to  the  alge- 
braic sum  of  the  elements  of  its  kth.  row  multiplied  by  their  minors, 
the  products  being  alternately  plus  and  minus  when  h  is  odd,  and 
alternately  minus  and  plus  when  k  is  even.  From  this  the  law  of 
signs  as  given  in  the  theorem  at  once  follows. 


Ex. 


a\ 

bi 

Cl 

di 

(h 

b. 

C2 

d. 

as 

b. 

C8 

ds 

a* 

64 

Ci 

di 

ai 

61 

Cl 

di 

02 

ft2 

C2 

d.2 

a* 

h 

Ci 

d4 

as 

63 

ca 

da 

61  Cl  di 

04  62  C2  di 

1 63  C8  da 


+  64 


«x 

61 

Cl 

di 

Oi 

64 

Ci 

d* 

02 

62 

C2 

do 

as 

63 

C3 

ds 

ai  Cl  di 
^2  C2  d^ 
as  C3  da 


-Ci 


at 

64 

C4 

di 

ai 

61 

Cl 

di 

02 

62 

C2 

d2 

as 

63 

C3 

ds 

a\   61  di 

02  62  d2 

03  63  d3 


+  di 


ai  61  Cl 
tta  62  C2 
as  63  C3 


When  a  determinant  is  thus  expressed  it  is  said  to  be  expanded 
according  to  the  elements  of  the  kth  row.  We  shall  call  the 
coefficient  of  an  element  the  quantity  which  multiplies  it  in 
the  expansion. 

Then  the  coefficient  of  an  element  is  plus  or  minus  the  corre- 
sponding minor  according  as  the  nu7nber  of  the  row  added  to  the 
number  of  the  column  is  eveii  or  odd. 


PROPERTIES  OF  DETERMINANTS 


9 


The  coefficient  of  a^  shall  be  denoted  by  A^,  that  of  h^  by  B^, 
and  so  on.    Then 


=  b,B,  +  b,B,+  b,B, 
=  c^C^  +  CgCg  +  C3C3 

=  a^A^+\B^+c^C^ 
=  a^A^+\B^+c^C^. 


4.  If  any  two  rows  {or  columns)  of  a  determinant  are  inter- 
changed, the  sign  of  the  determinant  is  changed. 

For  suppose  the  determinant  is  expanded  as  in  theorem  3,  and 
that  two  rows  other  than  that  used  in  the  expansion  be  inter- 
changed. A  similar  interchange  takes  place  in  the  minors  of  the 
expansion.  Hence,  if  the  theorem  is  true  for  each  of  the  minors, 
it  is  true  for  the  determinant.  In  other  words,  if  the  theorem  is 
true  for  determinants  of  any  order,  it  is  true  for  those  of  the  next 
higher  order.  But  the  theorem  is  certainly  true  for  determinants 
of  the  second  order.    Hence  it  is  always  true. 

5.  If  two  rows  (or  cohtmns)  of  a  determinant  are  the  same,  the 
determinant  is  equal  to  zero. 

Let  a  determinant  with  two  rows  the  same  be  expanded  accord- 
ing to  the  elements  of  some  other  row.  Each  minor  of  the  expan- 
sion has  two  rows  the  same.  Hence,  if  the  theorem  is  true  for 
determinants  of  any  order,  it  is  true  for  determinants  of  the  next 
higher  order.  But  the  theorem  is  certainly  true  for  determinants 
of  the  second  order,  for 


afi^- 


^1^1  ~  ^• 


Hence  it  is  universally  true. 


10 


ELIMINATION 


6.  Tlie  sum  of  the  products  obtained  by  multiplying  the  elements 
of  any  row  {or  column)  by  the  coefficients  of  the  corresponding  ele- 
ments of  some  other  row  {or  column)  is  zero. 

Consider,  for  example, 


=  a,A,  +  b,B^  +c^C,+  d,D^ 


If  we  replace  a^,  b^,  c^,  d^,  on  the  right-hand  side  of  this  equation 
by  a^,  b^,  c^,  d^,  the  same  substitution  must  be  made  on  the  left- 
hand  side.    Then  we  have 


=  a^A^  +  b^B^  +  c^C^  +  dj)^. 


«1 

\ 

H 

d 

«2 

K 

^2 

d, 

«S 

\ 

^3 

d, 

a. 

K 

^4 

d^ 

«i 

\ 

<^X 

d. 

«4 

h 

^4 

< 

«3 

h 

«3 

d. 

a. 

K 

'^4 

< 

But  the  determinant  is  zero,  by  theorem  5  ;  therefore 
a^A^+b,B^+  cf^+  d^D^  =  Q. 

It  is  evident  that   the  proof  is  general  and   establishes   the 
theorem. 

7.  If  each  element  of  any  row  {or  column)  is  multiplied  by  the 
same  quantity,  the  determinant  is  multiplied  by  the  same  quantity. 

This  follows  at  once  from  theorem  3.    For  example, 


8     h 


kc^ 

d. 

kc^ 

d. 

kc^ 

ds 

kc^ 

d. 

=  kc^  C^  +  kc^  C^  -f-  kc^  C3  +  kc^  (7^ 


=  kic^c,+ c,c^^+  c,c,+  c^c;\ 


«1 

K 

^1 

d. 

«2 

b. 

c. 

d. 

«3 

h 

^3 

d. 

«4 

h 

^4 

d. 

PROPERTIES  OF  DETERMINANTS 


11 


8.  If  each  of  the  elements  of  any  row  {or  column)  is  increased 
hy  the  same  multiple  of  the  corresponding  element  of  any  other 
row  {or  column),  the  value  of  the  determinant  is  unchanged. 

We  wish  to  show,  for  example,  that 


(1) 


r 


«1  \ 

h       ^1 

«2        h       ^1       ^2 

^3        ^3        Cg       d^ 

*4        ^4        ^4       ^4 

a^    b^  +  kd^ 

c,    d^ 

a^    \+M^ 

^2       ^2 

«3    &3+K 

^3    dz 

a^    h^+kd^ 

c,    d. 

(2) 


Let  the  coefficients  of  the  elements  in  the  second  column  of 
(1)  be  B^,  B^,  B^,  B^.  It  is  evident  that  these  are  also  the  coeffi- 
cients of  the  elements  of  the  second  column  of  (2).   Hence  (2)  is 

{\  +  kd;)  B^  +  {b^  +  kd^)  B^  +  (&3  +  Mg)  B^  +  (&,  +  kd^  B^, 

which  equals 

b^B^  +  b^B^  +  b^B^  +  b,B,  +  k  {d^B^  +  c?2^2  +  d^B^  +  d^B^). 

The  coefficient  of  k  in  this  equation  is  zero,  by  theorem  6,  and 
the  remaining  terms  equal  the  determinant  (1).    Hence  (2)  =  (1). 

It  is  evident  that  the  proof  is  general.  The  following  are  special 
cases :  If  A;  =  1,  the  elements  of  one  row  or  column  are  added  to 
the  corresponding  elements  of  another  row  or  column  ;  if  ^^  =  —  1, 
the  elements  of  one  row  or  column  are  subtracted  from  those  of 
another  row  or  column. 

This  theorem  is  often  used  in  simplifying  determinants. 


Ex.  1.   Consider 


1-2  1         2 

3-5  3         5 

-  1         2  3-4 

3-5  2          5 


(1) 


If  the  elements  of  the  second  cohimn  are  added  to  those  of  the  fourth  column 
this  becomes 

1-2      1         0 

3-5     3         0 

-1         2     3-2 

8-5     2         0 


(2) 


12 


ELIMINATION 


If  twice  the  elements  of  the  first  column  are  added  to  those  of  the  second 
column,  (2)  becomes 

10     1         0 

3     13         0  (3) 

-10     3-2-  ^  ' 

3      12         0 

If  the  elements  of  the  first  column  are  subtracted  from  those  of  the  third 
column,  (3)  becomes 

10         0         0 

3     10         0 

-10         4-2 

3     1-1         0 


(4) 


Expressing  (4)  as  the  sum  of  the  product  of  the  elements  of  the  first  row 
and  their  coefficients,  it  becomes 


and  this  is  equal  to 


1         0 
0         4     - 
1-1 

14-21 
-  1         0 


=  -2. 


Ex,  2.    Consider 


X 

y    1 

Xi 

yi    1 

X2 

2/2      1 

By  successive  subtraction  of  the  elements  of  one  row  from  those  of  another 
we  have 


X     y     I 

xi    yi    1 

= 

«2      2/2      1 

x-Xi     2/  -  2/1    0 
Xi  ^1  1 

^2  2/2  1 


X  -Xi     2/    -  2/1     0 
SCl  -X2     yi  -  2/2     0 

X2  2/2  1 


\x       Xi    2/       2/1      ^ijg  |f^g{  transformation  being   made  by 
X1-X2    2/1-2/2  ',  „  * 

theorem  3. 


4.  Solution  of  n  linear  equations  containing  n  unknown  quan- 
tities, when  the  determinant  of  the  coefficients  of  the  unknown 
quantities  is  not  zero.  We  are  now  prepared  to  show  that  the 
method  used  in  §  1  to  solve  equations  with  two  or  three  unknown 
quantities  can  be  so  generalized  as  to  apply  to  any  system  of 
equations  of  the  first  degree  in  which  the  number  of  equations  is 
equal  to  the  number  of  the  unknown  quantities.  For  convenience 
we  will  take  the  case  of  four  equations,  but  the  student  will  readily 
see  that  the  method  is  perfectly  general. 


n  EQUATIONS  WITH  n  UNKNOWNS 

Cousider  the  equations 

a^x  +  bji/  +  c^z  +  djW  +  e^  =  0, 
a^x  +  b^y  +  c^  +  d^w  +  e^  =  0, 
a^x  +  h^y  +  C32  +  d^w  +  e^  =  0, 
a^a;  +  &42/  +  c^z  +  f^^w  +  e^  =  0. 


13 


(1) 
(2) 
(3) 
(4) 


«1 

^ 

^1 

^1 

a^ 

K 

^2 

^2 

«8 

h 

C3 

^3 

^4 

K 

^'4 

d. 

Let  the  determinant  of  the  coefficients  of  the  unknown  quan- 
tities X,  y,  z,  w  be  denoted  by  D,  so  that 


D  = 


and  let  A^  denote  the  coefficient  of  a^,  B^  the  coefficient  of  &^,  and 
so  on.    We  assume  i>  ^  0. 

If  now  we  multiply  (1)  by  A^,  (2)  by  ^„  (3)  by  A^,  (4)  by  A^, 
and  add  the  results,  we  have,  by  theorems  3  and  6,  §  3, 

Dx  +  c^A^  +  ^2^2  +  ^3^3  +  ^4^4  =  0-  (5) 

Similarly,  by  using  B^,  B^,  B^,  B^  as  multipliers,  we  have 

By  +  e,B^  +  e^B^  +  ^3^3  +  e^B^  =  0 ;  (6) 

by  using  C^,  C^,  C^,  C^  as  multipliers,  we  have 

Bz  +  e^C,+  e,C,+  e^C,+  e^C,  =  0;  (7) 

and  by  using  D^,  D^,  D^,  D^  as  multipliers,  we  have 

Dw  +  e^D^  +  ej)^  +  e3Z>3  +  ej)^  =  0.  (8) 

Now  it  is  clear  that  any  values  of  x,  y,  z,  w  which  satisfy  (1), 
(2),  (3),  (4)  satisfy  also  (5),  (6),  (7),  (8).  Conversely,  any  values 
which  satisfy  (5),  (6),  (7),  (8)  satisfy  also  (1),  (2),  (3),  (4).  For  if 
we  multiply  (5)  by  a^,  (6)  by  h^,  (7)  by  c^  (8)  by  d^,  and  add,  we 
obtain  (1).  Similarly  (2),  (3),  (4)  can  be  obtained  from  (5),  (6), 
(7),  (8).  Hence  (1),  (2),  (3),  (4)  and  (5),  (6),  (7),  (8)  are  equivalent 
equations. 


14 


ELIMINATION 


Now        e^A,  +  e^A^  +  e^A^  +  e^A^  = 


e,B^^e^B^-^e,B^  +  e,B,= 


e,C,+  e,C,+  e^C,+  e,C,= 


«, 

^ 

<^1 

d. 

^. 

K 

^2 

d. 

«« 

h 

^8 

< 

«4 

\ 

^4 

d. 

«1 

H 

'^l 

d. 

^2 

^2 

Cg 

d. 

«8 

«3 

H 

d. 

a. 

«4 

H 

d. 

tj    5j    e^    d^ 


«3       ^3       ^8       ^8 
*4       ^4      ^4       ^4 


and 


6,i),+  e,l>2+e3A+«4^4  = 


^1  ^1  ^1  ^1 

^2  62  c^  e^ 

^3  ^3  ^3  ^3 

^4  &4  C4  «^ 


Hence  the  solution  of  (5),  (6),  (7),  and  (8)  is 


X  — 


«1 

\ 

\ 

^. 

«« 

K 

«2 

«!, 

«« 

h 

<^3 

^8 

«4 

h 

<^, 

< 

a, 

\ 

^1 

d. 

«2 

K 

«2 

d. 

«8 

■K 

^3 

< 

^4 

h 

C4 

^4 

2/  = 


«1 

«1 

«1 

^X 

«2 

«2 

^2 

d. 

«3 

«3 

^8 

d. 

^4 

«4 

«4 

d. 

«1 

^ 

^1 

d. 

^2 

^2 

^2 

d„ 

a^ 

&3 

C3 

d. 

^4 

&4 

^4 

d. 

a. 

\ 

«i 

^X 

«2 

K 

«2 

«^2 

«8 

h 

«3 

«^8 

«4 

K 

«4 

< 

«t 

h 

^'l 

d. 

«2 

K 

^2 

d. 

«« 

K 

<^3 

d. 

«4 

h 

^4 

< 

«i 

^ 

Ci 

«1 

^2 

^0 

^2 

^2 

«3 

^3 

«3 

«3 

«4 

K 

C4 

«4 

«1 

K 

Cl 

^1 

«2 

K 

«2 

«^2 

«3 

h 

«8 

«58 

«4 

h 

C4 

d. 

and  this  is  the  solution,  and  the  only  solution,  of  (1),  (2),  (3),  (4). 


n  EQUATIONS  WITH  MORE  THAN  n  UNKNOWNS     15 


Hence  we  may  state  the  following  important  theorem : 

Any  system  of  n  linear  equations  containing  n  unknown  quan- 
tities has  one  and  only  one  solution  when  the  determinant  formed 
by  the  coejfficients  of  the  unknown  quantities  is  not  zero. 

This  solution  may  be  written  down  at  once,  for  each  unknown 
quantity  is  equal  to  mmus  a  fraction,  of  which  the  denominator  is 
the  determinant  of  the  coefficients  and  the  numerator  is  a  similar 
determinant  formed  by  replacing  the  coefficients  of  that  unknown 
quantity  by  the  absolute  terms. 

Ex.  1. 


Ex.  2. 


3x  +  52/-4  =  0, 

2a; -32/  +  7  =  0. 

-4 

5                                           3 

-4 

7 

-3           23                            2 

7 

29 

3 

6      "      19'             ^^       3 

6 

~  19 

2 

-3                                             2 

2x-3y+     z-l  =  0, 

4x  +  5y-22  +  2  =  0, 

x-2  7/ +  32-3  =  0. 

-3 

1-3  1 
2  5-2 
3-2         3 


2-3         1 

4         5-2 
1-2         3 


=  0, 


y  =  - 


2 

-3 

-1 

4 

5 

2 

1 

-2 

-3 

2 

-3 

1 

4 

5 

-2 

1 

-2 

3 

2-1  1 

4         2-2 
1-3         3 


-3         1 

5-2 
-2         3 


=  1. 


=  0, 


5.  Systems  of  n  linear  equations  containing  more  than  n  un- 
known quantities.  When  in  a  set  of  linear  equations  the  number 
of  equations  is  less  than  the  number  of  unknown  quantities,  the 
equations  have  usually  an  infinite  number  of  solutions,  but  may 
have  none.  The  general  method  of  procedure  in  solving  them  is  to 
pick  out  a  number  of  the  unknown  quantities  equal  to  the  number 
of  the  equations  and  having  the  determinant  of  their  coefficients 


IQ  ELIMINATION 

not  zero.    These  are  solved  by  the  method  of  §  4.    We  then  have 
these  unknown  quantities  expressed  in  terms  of  the  others. 

Ex.1.  2x  +  Sy+     2  +  4  =  0, 

x-2y  +  3z  +  2  =  0. 


I>  =  \'        ?l=-7. 


If  we  choose  x  and  y  for  the  unknown  quantities,  we  have 

12         31 
|l     -2| 

Then,  solving  as  in  §  4,  we  have 

I    z  +  4         31 
3z  +  2     -2  11 

a;  =  — 1 — -  = z  —  ^, 

2         3  7  ' 

|l     -2| 

12       z  +  4| 

1     3z  +  2       5 

«  =  —  —. — ^  =  -z : 

"  2         3  7 

|l     -2| 

and  since  z  may  be  given  any  value  whatever,  the  equations  have  an  infinite 
number  of  solutions. 

Ex.2.  2x  +  3y+     z  +  4  =  0, 

2x  +  3y  +  2z  +  3  =  0. 


If  we  choose  to  solve  for  x  and  y,  we  have 

12     3 
2     3 


I)  =  \l     f|  =  0. 


But  if  we  choose  to  solve  for  y  and  z,  we  have 

n—         ^  —  ^ 

-^-13     2|-'^- 

The  solutions  are  y  =  —  ^x  —  ^, 

z  =  l. 

It  is  possible  that  no  selection  of  the  unknown  quantities  will 
lead  to  a  determmaut  of  the  coefficients  which  is  not  zero.  In  this 
case  the  equations  may  have  no  solution.  The  discussion  is  too 
complex  for  this  book,  but  the  student  will  probably  have  no  diffi- 
culty with  the  cases  likely  to  occur  in  practice. 


n  EQUATIONS  WITH  n  UNKNOWNS  17 

Ex.3.  2x  +  Sy  +  z  +  4  =  0, 

2x  +  3y  +  z  +  S  =  0. 

The  determinant  of  any  pair  of  unknown  equations  is  zero.  By  subtracting 
the  second  equation  from  the  first  we  have  1  =  0,  showing  the  equations  to  be 
contradictory. 

6.  Systems  of  n  linear  equations  containing  n  unknown  quan- 
tities, when  the  determinant  of  the  coefficients  of  the  unknown 
quantities  is  zero.  Consider  again  equations  (1),  (2),  (3),  (4)  of 
§  4,  but  with  the  assumption  that  Z>  =  0.  We  may  proceed  exactly 
as  in  §  4,  but  equations  (5),  (6),  (7),  (8)  do  not  now  contain  the 
unknown  quantities.  In  fact,  these  equations  are,  in  general,  con- 
tradictory, and  consequently  equations  (1),  (2),  (3),  (4)  have,  in 
general,  no  solution, 

Ex.  1.  X  -  2/  +     z  +  3  =  0, 

2x-{-y  +  Sz  +  1  =  0, 
«  +  2y  +  2z  +  4  =  0. 


Here  B 


1-11 
2  13 
1         2     2 


0. 


Eliminating  y  and  z  by  the  method  of  §  4,  we  have  0  x  —  24  =  0,  which  is 
absurd.    Hence  the  equations  have  no  solution. 

It  is,  of  course,  possible  that  when  D  =  0  each  of  the  other 
determinants  in  (5),  (6),  (7),  (8)  Ib  also  zero.  Each  of  these  equa- 
tions is  then  simply  0  =  0,  and  gives  no  direct  information  about 
the  solutions  of  (1),  (2),  (3),  (4).  As  a  matter  of  fact,  in  this  case, 
(1),  (2),  (3),  (4)  have,  in  general,  an  infinite  number  of  solutions, 
but  may,  under  special  conditions,  have  no  solutions. 

The  general  discussion  is  too  complex  to  be  given  here.  We 
shall  simply  state  the  following  theorem : 

A  set  of  linear  equations  containing  n  unknown  quantities  has, 
in  general,  no  solution  when  the  determinant  of  the  coefficients  of 
the  unknoivn  quMntities  is  zero,  but  may,  under  certain  conditions, 
have  an  infinite  number  of  solutions. 


13  ELIMINATION 

In  practice,  one  of  the  n  equations  may  be  temporarily  set  aside, 
and  the  other  w  —  1  equations,  which  contain  11  unknown  quan- 
tities, may  be  examined  by  the  method  of  §  5.  If  these  equations 
can  be  solved,  the  solution  can  be  tested  in  the  equation  which 
has  been  set  aside. 

Ex.  2.  2x-    32/+     2-1  =  0, 

X-    22/  +  3z  +  4  =  0, 

7a;-ll?/  +  6z+ 1  =  0. 

If  the  method  of  §  4  is  used,  the  result  is  0  =  0.  Solving  the  first  two  equa- 
tions for  X  and  y,  we  have 

ic  =  7  z  +  14, 
y  =  5  z  +    9, 

and  these  results  are  found  on  trial  to  satisfy  the  last  of  the  given  equations. 
Since  z  may  have  any  value,  the  equations  have  an  infinite  number  of  solutions. 


7.  Systems  of  linear  equations  in  which  the  number  of  the 
equations  is  greater  than  that  of  the  unknown  quantities.  If 
there  are  more  equations  of  the  first  degree  than  there  are  unknown 
quantities,  there  will  be,  in  general,  no  values  of  the  imknown 
quantities  which  satisfy  all  equations.  There  may  be  such  values, 
however,  when  certain  relations  exist  among  the  coefficients  of 
the  equations.  To  obtain  these  relations  we  may  pick  out  a  num- 
ber of  equations  equal  to  the  number  of  tlie  unknown  quantities 
and  solve  them.  If  the  solution  is  substituted  in  the  remainmg 
equations,  there  will  result  certain  expressions  in  the  coefficients 
which  must  be  zero  if  the  equations  are  to  be  satisfied. 

The  most  important  case  is  that  in  which  there  are  n-\-l  equa- 
tions containing  n  unknown  quantities.    For  example,  consider 

a^x  +  l^y  -f-  Cj2  +  d^  =  Q, 
a^x  +  hju  +  c^  +  rf.,  =  0, 
a^x  -h  &^  -h  c„z  +  <3  =  0, 
a, A'  -\-'^i^y  +  c^z  +  (l^  =  0. 


n  +  1  EQUATIONS  WITH  n  UNKNOWNS 


19 


The  solution  of  the  first  three  equations,  if 


^0,  is  (§4) 


x  = 


d, 

^ 

<^X 

d. 

(>. 

^2 

ds 

K 

^3 

«i 

\ 

'^l 

«2 

K 

«2 

«3 

^3 

^3 

^ 

^'i 

^1 

*2 

^2 

«!, 

*3 

^3 

d. 

«1 

K 

^1 

a^ 

^2 

.«2 

«3 

^3 

«3 

«1 

d. 

^1 

«2 

d. 

<^2 

«3 

ds 

C3 

«1 

\ 

«1 

«2 

b. 

^2 

«3 

^ 

C3 

«1 

«1 

^. 

«2 

^2 

d. 

«3 

^3 

d. 

«1 

^ 

h 

«2 

*2 

Cg 

«3 

^ 

«3 

«1 

^ 

^, 

«2 

62 

^2 

«3 

*3 

«53 

«1 

^'l 

«1 

«2 

K 

^2 

«3 

*3 

^3 

Substituting  these  values  in  the  first  member  of  the  last  equa- 
tion, we  have 


—  a. 


h    ^1 


+  K 


\    d^ 
".    d^ 


—  c. 


a^    \    d^ 

^2     h     ^2 

+  d. 

«3       ^3       ^3 

b,    c, 

\       ^2 


«2       K       ^2 


which,  by  theorem  3,  §  3,  is  the  same  as 


h 

«i 

^^ 

K 

^2 

^^2 

\ 

C3 

d. 

b. 

^4 

d. 

^2       «2 


20 


ELIMINATION 


a, 

\ 

h 

rf 

«2 

K 

^2 

d. 

^8 

h 

^3 

d. 

«4 

h 

^4 

d 

Hence,  in  order  that  the  last  equation  may  be  satisfied,  we  must 
have 


=  0. 


Extending  this  to  any  number  of  variables,  we  have  the  theorem : 

In  order  that  a  system  of  n  +  1  linear  equations  containing  n 
unknovm  quantities  shall  have  a  solution,  it  is  necessary  that  the 
determinant  formed  from  the  coefficients  of  the  unknown  quantities 
and  the  absolute  terms  shall  he  zero. 


Ex.  1. 


x+  2/+  z-2  =  0, 
2x+     y-    z  +  3  =  0, 

x-22/-3«  +  4  =  0, 
5x-3y-4«  +  l  =  0. 


Here 


111-2 
2  1-1  3 
1  _2  -3  4 
6-3-4         1 


=  0, 


showing  that  if  the  first  three  equations  have  a  solution  it  will  satisfy  the  fourth 
equation.    In  fact,  the  solution  is  x  =  1,  y  =  —  2,  z  =  3. 

It  should  be  noted  that  the  converse  of  the  theorem  stated  is 
not  necessarily  true.  All  that  has  been  proved  is  that  if  n  of  the 
equations  have  a  solution,  that  solution  satisfies  the  {n  +  l)st  equa- 
tion when  the  determinant  is  zero.  But  the  determinant  may  be 
zero  when  the  equations  are  contradictory. 


Ex.2. 


2x-3?/+  2  +  1  =  0, 
2x-3y +  52  +  2  =  0, 
2x-3i/-6z-3  =  0, 
2x-3?y  +  2z-8  =  0. 


Here 


2-311 
2-362 
2  -3  -6  -3 
2-3         2-8 


=  0, 


but  any  three  of  the  equations  may  be  seen  to  be  contradictory  by  the  method 
of  §8. 


LINEAR  HOMOGENEOUS  EQUATIONS  21 

8.  Linear  homogeneous  equations.  An  equation  is  homoge- 
neous with  respect  to  the  unknown  quantities  when  the  sum  of 
the  exponents  of  the  unknown  quantities  is  the  same  in  each  term. 
In  particular  an  equation  of  the  first  degree  is  homogeneous  when 
each  of  the  terms  contains  one  of  the  unknown  quantities;  for 
example, 

where  x^,  x^,  x^,  x^  are  the  unknown  quantities. 

Tliis  equation  is,  of  course,  satisfied  by  placing  x^  =  0,  x^  =  0, 
x^  =  0,  x^  =  0,  but  in  practice  tliis  solution  is  generally  unimportant. 
In  such  equations,  in  fact,  it  is  usually  the  ratios  of  the  unknown 
quantities  wliich  are  important ;  for  if  each  unknown  quantity  is 
multiplied  by  the  same  number,  the  equation  is  unaltered.  In  fact, 
if  we  place 

X, 


X 


1  "^2  "^3 


the  homogeneous  equation  just  written  becomes  the  non-homogene- 
ous equation 

a^  -H  a^  +  a^  +  a^  =  0. 

In  this  manner  a  set  of  homogeneous  equations  containing  n 
unknown  quantities  may  be  reduced  to  a  set  of  non-homogeneous 
equations  containmg  ii  —  \  unknown  quantities  by  dividing  each 
equation  by  one  of  the  unknown  quantities.  The  methods  of  the 
previous  articles  may  then  be  used.  But  tliis  method  of  proced- 
ure is  open  to  the  objection  that  the  unknown  quantity  by  which 
the  equations  are  divided  may  possibly  be  zero  when  the  division 
is  invalid.  It  is  better,  therefore,  to  handle  the  homogeneous  equa- 
tions as  they  stand,  slightly  modifying  the  methods  used  for  non- 
homogeneous  equations  in  a  manner  which  will  be  clear  from  the 
examples. 

Ex.  1,  a^xx  +  a'^X'!,  +  03X3  +  ^4X4  =  0, 

61X1  +  62X2  +  63X3  +  &4X4  =  0,  (1) 

C]X\  +  C2X2  +  CgXs  +  C4X4  =  0. 


'>') 


ELIMINATION 


We  will  handle  this  by  the  method  of  §  4,  in  that  we  temporarily  look  upon 
Xi,  Xa,  Xa  as  the  unknown  quantities.    We  have,  in  the  first  place, 


=  0, 


61 

02 
62 

63  Xi  + 

O4X4  fl2  O3 
64X4       62       &3 

Cl 

C2 

C3 

C4X4       C2       Cs 

61 

02 
62 

03 
63 

X2  + 

Oj  04X4  03 
6i       64X4       63 

Cl 

C2 

C3 

Cl        C4X4       C3 

ai 
61 

O2 
62 

03 
63 

X8  + 

Oi  C2  O4X4 
61       62       64X4 

Cl 

Ca 

cs 

Cl       C2       C4X4 

=  0, 


=  0, 


which  may  be  written  as 


Oi 

02 

03 

61 

fe2 

&3 

Xi  + 

Cl 

C2 

C3 

Oi 

02 

as 

h 

62 

63 

X2- 

Cl 

C2 

C3                 1 

Ox 

O2 

O3 

61 

62 

63 

X3  + 

Cl 

C2 

Cs 

O2 

as 

04 

62 

hs 

64 

C2 

Cs 

C4 

fli 
61 

Oa 

&3 

O4 
64 

Cl 

Cs 

C4 

Oi 

02 

62 

O4 

hi 

Cl 

C2 

Ci 

X4  =  0, 


X4  =  0, 


X4  =  0. 


(2) 
(3) 
(4) 


From  these  follow : 


Xi :  X2  :  Xs  :  X4  = 


02 

O3 

04 

62 

63 

64 

:  — 

C2 

Cs 

C4 

Oi 

03 

04 

1 

61 

63 

64 

•  1 

Cl 

Cs 

C4 

1 

Oi 

02 

O4 

&1 

62 

h 

:  — 

Cl 

C2 

C4 

ai 

02 

as 

61 

62 

63 

Cl 

C2 

Cs 

(5) 


The  result  (5)  holds  even  when  one  or  more,  but  not  all,  of  the  determinants 
involved  are  equal  to  zero.  Then  the  corresponding  unknown  quantities  are 
equal  to  zero.    For  example,  if 


Oi 

03 

04 

61 

h 

64 

=  0, 

Cl 

Cs 

C4 

Ol 

02 

04 

61 

62 

h 

Cl 

C2 

C4 

and  the  other  determinants  in  (5)  are  not  zero,  (3)  and  (4)  show  that  X2  =  0  and 
Xs  =  0,  while  (2)  shows  that  the  ratio  of  Xi  and  X4  are  correctly  given  by  (5). 

If  all  the  determinants  in  (5)  are  zero,  the  values  of  the  unknowns  are  not 
thereby  determined.  In  this  case,  two  of  the  equations  (1)  should  be  solved  for 
two  of  the  unknown  quantities  in  terms  of  the  others,  and  the  results  tested  for 
the  last  equations. 


ELIM^ANTS  23 

It  should  be  noted  that  contradictory  equations  cannot  occur.    The  student 
should  compare  the  contradictory  equations 

2x-3y  +  4  =  0, 

2x-3y  -2=0, 

with  the  homogeneous  equations 

2xi  —  3iC2+4  Xs  =  0, 
2  »i  —  3  a^  —  2  ajs  =  0. 

By  subtracting  one  equation  from  the  other  we  have 

6x3  =  0, 
whence  X3  =  0  and  Xi :  X2  =  3 : 2. 

Ex.  2.  The  four  equations 

aiXi  +  aix<2.  +  03X3  +  04X4  =  0, 
hyxx  +  623:2  +  63X3  +  643:4  =  0, 
CiXi  +  C2X2  +  C3X3  +  C4X4  =  0,    . 
d\Xx  +  ^23:2  +  ^3X3  +  ^4X4  =  0, 

have,  of  course,  the  common  solutious,  xi  =  0,  X2  =  0,  X3  =  0,  X4  =  0.  In  order 
that  they  may  also  be  satisfied  by  the  same  ratios  of  the  unknown  quantities,  it 
is  necessary  that 

a\    a<i,    as    04 

61     62     63     64 

C\      Ci      C3      C4 

d\    d^    ds    (^4 


=  0. 


The  proof  is  as  in  §  7.  The  condition  is  also  sufiBcient,  for  the  proof  of  §  7 
shows  that  if  three  of  the  equations  have  a  solution,  that  will  also  be  a  solution 
of  the  fourth  equation  ;  and,  as  just  noted,  three  homogeneous  equations  always 
have  a  solution. 

9.  Eliminants.  The  result  of  eliminating  all  the  unknown 
quantities  from  two  or  more  equations  is  an  equation  the  left- 
hand  member  of  wliich  is  called  the  eliminant,  or  resultant,  of 
the  given  equations.     The  following  cases  are  important: 

1.  w  + 1  non-homogeneous  linear  equations  with  n  unknown 
quantities.  To  eliminate  the  unknown  quantities,  we  may  solve 
n  of  the  equations  and  substitute  the  solutions  in  the  remaining 
equation.    The  work  and  the  result  are  as  in  §  7 ;  that  is, 

The  eliminant  of  n  +  1  non-homogeneous  equations  with  n  utv- 
known  quantities  is  equal  to  the  determinant  of  the  coefficients  and 
the  absolute  terms. 


24  ELIMINATION 

2.  n  homogeneous  linear  equations  with  n  unknown  quantities. 
To  eliminate  the  unknown  quantities,  we  may  solve  n-1  equa- 
tions for  their  ratios  and  substitute  the  results  in  the  remaining 
equation.    The  work  and  the  result  are  as  in  §  8 ;  that  is, 

The  eliminant  of  n  homogeneous  equatwns  with  n  unknown  quan- 
tities is  equal  to  the  determinant  of  the  coefficients. 

3.  Two  equations  containing  one  unknown  quantity.  Let  it  be 
required  to  eliminate  x  between  the  equations 

a,ar^+M  +  ^i  =  ^'  (^) 

a,.r^+6,a^  +  C3  =  0.  (2) 

If  we  multiply  each  equation  by  x,  we  have 

a^n?  +  h^a?'  +  c^x  =  0,  (3) 

and  a^x^  +  h^a?  +  c„x  =  Q.  (4) 

These  four  equations  may  now  be  considered  as  Hnear  in  the 
three  unknown  quantities  a?,  yp-,  and  x.    Elimination  gives,  by  1, 


=  0.  (5) 


It  is  clear  that  if  equations  (1)  and  (2)  have  a  common  solution, 
equation  (5)  must  be  true.  Conversely,  it  may  be  shown  that  if  (5) 
is  true,  (1)  and  (2)  must  have  a  common  solution  ;  but  this  proof  is 
too  long  to  be  given  here. 

The  method  used  in  the  above  problem  may  be  used  for 
equations  of  any  degree  and  is  known  as  Sylvester's  method  of 
elimination.  It  consists  in  multiplying  the  given  equations  by 
successive  powers  of  x  until  we  have  one  more  equation  than  we 
have  powers  of  x.    The  eliminant  is  then  found  as  in  1. 

The  method  may  also  be  used  to  eliminate  one  of  the  unknown 
quantities  from  two  equations  containing  two  unknown  quantities. 


0 

«i 

\ 

^l 

0 

«2 

K 

^2 

«1 

\ 

^1 

0 

O-c, 

K 

Co 

0 

PROBLEMS 


25 


PROBLEMS 


Find  the  value  of  each  of  the  following  determinants  i 


1. 


4. 


14     51 
3    6 


9  \^ 

^'  \y 

n  IX 

^-  X2 


I  1   o| 

[10     4|" 

II  -71 
|2         3| 

12    3 

2  3     1 

3  12 


7. 


10. 


1 

0     1 

3 

2     0 

1 

1     0 

0 

a    b 

a 

0    c 

b 

c     0 

a 

h    g 

h 

b    f 

9 

f    c 

1 

1      1 

a 

b     c 

a2 

62 

C2| 

11. 


12. 


13. 


X  y  1 
12-1 
4     3         2 


1 

1 

1 

0 

0 

1 

1 

1 

1 

0 

1 

1 

1 

1 

0 

1 

0  ai  bi  Ci 

0  ag  62  C2 

«!  bi  ci  0 

02  62  C2  0 


Prove  the  following  relations  ; 


14. 


15. 


16. 


20. 


21. 


22. 


23. 


4     2     1 

1 

-2 

-3 

1    2    3 

3     4     2 

=  0. 

17. 

-2 

1 

3 

= 

2     1    3 

6    6     3 

-3 

1 

2 

3     1    2 

5     4     1 

3     2     1 

=  0. 

X     y 

z 

1     1     1 

2     1     1 

18. 

x2    y2 

Z2 

=  xyz 

xyz 

X8       w3 

Z8 

x2     2/2     z'^ 

a    0    0    6     0 

0    a    0    0     b 

X    y    I    z    w 

=  (ad- 

-6c)2 

1         1 

1 

c     0    0    d    0 

19. 

X     y 

Z 

=  {x-y){y-z){i 

0     c     0    0     d 

X2      2/2 

22 

ai    61    0      0 

a2    62    0     0 

_  ffli     &i 

Cl 

di. 

0      0     ci    di 

02    ^2 

C2 

da 

0         0        C2      ^2 

1         4-3 

6 

5 

0 

2         1 

0         6-8 

-  1 

1     - 

-6 

-3         4 

2-3         4 

2 

— 

3 

8 

4     -3 

5         1 

3 

4 

4 

1 

2 

5 

«!  +  di  6i 
a2  +  d2  bo 
as  +  dz    63 


«!  61 
a2  62 
as     63 


ao  3ai  3a2  as  0 

0  ao  3ai  3a2  as 

ao  2ai  a2  0  0 

0  ao  2ai  02  0 


0 


0 


ao  2  oi  a2 


ao 


di    61 

Cl 

d2    62 

ca 

ds    63 

cs 

ai  2a2  as  0 

0  tti  2a2  as 

ao  2  ai  a2  0 

0  ao  2ai  ao 


ao  {a^a^  -  G  aoajaoas  +  4  aoa|  +  4  a^as  -  3  af aa^}. 


26  ELIMINATION 

Solve  the  following  equations : 


24. 


|4-x      3 
3      d-x 


=  0. 


25. 


1-x       2 

2  3-x 

3  5 


3 

6 

—  X 


=  0. 


Write  the  following  equations  in  their  expanded  forms : 


26. 


27. 


X        y    \ 

1 

X2  +  2/2      X 

y 

1 

2-3     1 

= 

0. 

28. 

5          1 

2 

1 

1         6     1 

13         2 

-3 

1 

2      -1 

-1 

1 

x^  +  y^  X    y 

1 

1        0     1 
1         1     0 
0        0    0 

1 
1 
1 

=  0. 

29. 

a  —  X      h 
h      b  —  X 

=  0. 

a 

—  X 

h 

9 

30. 

h 

b-x 

f 

=  0. 

9 

f 

C  —  X 

=  0. 


Solve  the  following  equations ; 

31.  4x-5y  +    0  =  0, 
7x-9y  +  11  =  0. 

32.  X  +  2  y  -    2  +  3  =  0, 
2x-    y  -5  =  0, 

X  +22-8  =  0. 


y     z 

1     1     . 

-  +  -  =  4. 
z      X 


34.  2x+    4y +  32  -  2  =  0, 

X-    5j/+     2+1  =  0, 
3x+  lOy  +  52  -  5  =  0. 

35.  2x+  y+  2  +  2  =  0, 
6x  +  2y  +  32  +  6  =  0, 
2x  +  3y-22  +  2  =  0. 

36.  X +  2/ +  92 -7  =  0, 
5x  -  y  +  fi2  -  5  =  0, 
3x- j/  +  3a-2  =  0. 


37.  10x-3y +  122-  5  =  0, 

4x-     y+    62-3  =  0, 
5x-2y  +    32         =0. 

38.  X  +  y  +  z  -  a, 
y  +  z  +  w  =  b, 

2  +  10  +  X  =  C, 

w  +  X  +  y  =  d. 

39.  lOxi  +  4x2  +  6x3  =  0, 

3xi+    X2  + 2x3  =  0. 

40.  xi  +  5x2 +  3x3  =  0, 

3  xi  +  3  X2  +    X3  =  0. 

41.  2Xi  +  4X2  +  X3  =  0, 

3xi  +  6x2  -X3  =  0. 

42.  2xi+  X2-5X3+  X4  =  0, 
3xi  -2x2  -  4x3  -  2x4  =  0, 

«i  +    X2  +  2  X3  -    X4  =  0. 

43.  2xi  -  3x2  +  2x3  -  3x4  =  0, 
4xi  +  5x2  +  4x3  -  6x4  =  0, 
3  xi  -  7  X2  -  2  X3  +  3  X4  =  0. 

44.  7xi  -  5x2  +  3x3  -  4x4  =  0, 
3xi+  2x2-  6x3  +  9X4  =  0, 
6X1  -  16  xo  +  21 X3  -  35  X4  =  0. 


PROBLEMS 


27 


Find  whether  or  not  the  equations  in  each  of  the  following  examples  have 
a  common  solution : 


45.  2x  -  2/  +  3  =  0, 
3a;  +  y-  1  =  0, 
3x-4y  +  10  =  0. 


46. 


6X-2  2/  +  7  =  0, 

3a;-    y  +  6  =  0, 

x  +  32/-l  =  0. 


47.  X-    2y+    3z-  1  =  0, 
2x+      y-      2  +  1  =  0, 

X-    3?/+    2z  +  2  =  0, 
X  -  19  y  +  22  2  -  4  =  0. 

48.  X-22/+  1  =  0, 
2/-2z  +  2  =  0, 
2-2x  +  3  =  0, 

X  +  ?/  +  2  =  0. 


49.  For  what  values  of  a  are  the 
following  equations  consistent  ? 

X  +  a'h)  +  a  =  0, 
ox  +  y  +  a2  =  0, 
a^x  +  ay  +  1  =  0. 

50.  Eliminate  x  from  the  equations 

X2/  +  3  X  +  1  =  0, 
2x2/ -4?/ +  2  =  0. 

51.  Eliminate  x  from  the  equations 

xy2  +  22/  +  3  =  0, 
xj/  +  4  X  +  1  =  0. 

52.  Eliminate  x  and  2  from   the 
equations 

xy  +  2/z  —  X  +  2  +  2  =  0, 

X2/-2x  +  y  +  2  +  2  =  0, 

X  +  3Z-2  =0. 


53.  Find  the  condition  that 

ax^  +  6x  +  c  =  0, 
and  x2  =  1, 

have  a  common  root. 

54.  Show  that  the  condition  that 

ax2  +  6x  +  c  =  0, 
and  x^  =  1, 

have  a  common  root  is 


=  0. 


55,  Show  that  if 

flix  +  hiy  +  ci  =  0, 

a^x  +  hiv  +  C2  =  0, 

asx  +  ftsy  +  C3  =  0, 

have  a  common  solution,  there  can 

always  be  found  three  numbere  Z,  A:, 

?n,  such  that 

«li  +  ^2^  +  «3"i  =  0» 

61/  +  62^  +  ftsWi-  =  0, 
Cii  +  C2^  +  Cz>n  =  0. 


a 

6 

c 

6 

c 

a 

c 

a 

b 

CHAPTEE   II 
GRAPHICAL  REPRESENTATION 

10.  Real  number.  The  science  of  mathematics  deals  with  vari- 
ous kinds  of  numbers,  each  of  which  has  arisen  through  the  desire 
to  perform,  without  restriction,  some  one  of  the  fundamental  oper- 
ations. The  simplest  numbers  are  the  positive  integers,  or  whole 
numbers.  If  one  restricts  himself  to  the  use  of  these,  he  may  add 
or  multiply  together  any  two  of  them  without  obtaining  a  new 
kind  of  number ;  but  he  may  not  divide  one  number  by  another  not 
exactly  contained  in  it,  nor  subtract  a  larger  number  from  a  smaller. 
In  order  that  division  may  always  be  performed,  the  common  frac- 
tions, which  are  the  quotients  of  one  integer  divided  by  another, 
are  necessary.  In  order  that  subtraction  may  always  be  possible, 
the  idea  of  a  negative  number  must  be  introduced.  The  integers 
and  fractions,  both  positive  and  negative,  together  form  the  class 
of  rational  numbers.  On  these  numbers  the  operations  of  addition, 
subtraction,  multiplication,  and  division  may  always  be  performed 
without  leading  to  a  new  kind  of  number. 

The  operation  of  evolution,  however,  leads  to  two  new  kinds  of 
numbers,  —  the  irrational,  exemplified  by  V2  ;  and  the  complex,  of 
wliich  V—  2  is  an  example.  The  complex  numbers  will  be  noticed 
in  §  12 ;  we  shall  here  speak  only  of  the  irrational  numbers.  An 
irrational  number  is  defined  as  one  wliich  cannot  be  expressed 
exactly  as  an  integer  or  a  common  fraction,  but  which  may  be  so 
expressed  approximately  to  any  required  degree  of  accuracy.  The 
simplest  examples  are  the  roots  of  rational  numbers ;  for  example, 
V?  may  be  approximately  expressed  as  f f^^  ro-Uh  ^^^■'  ^^^^  c^^" 
not  be  expressed  exactly.  There  are  also  irrational  numbers  which 
are  not  the  roots  of  numbers  and  cannot  be  expressed  by  means  of 
radical  signs.  A  familiar  exj^mple  is  the  number  7r  =  3.14159---. 
An  irrational  number  may  be  either  positive  or  negative.     The 

28 


ZEKO  AND  INFINITY  29 

rational  and  the  irrational  numbers  together  form  the  class  of  real 
numbers. 

A  rigorous  investigation  of  the  nature  and  properties  of  these 
numbers,  especially  of  the  irrational  numbers,  is  too  advanced  for 
this  book.  An  elementary  discussion,  however,  is  given  in  any 
course  in  algebra,  and  is  here  assumed  as  known. 

The  real  numbers  may  be  represented  graphically  on  a  number 
scale,  constructed  as  follows : 

On  any   straight  line  assume  a      — j — i — i — i — i — i — i — i-i-f 

•^  *^  -U  -3  -2    -1      0      1      2      S      U 

fixed  point  0  as  the  zero  point,  or 
origin,  and  lay  off  positive  numbers 

in  one  direction  and  negative  numbers  in  the  other.  If  the  line 
is  horizontal,  as  in  fig.  1,  it  is  usual,  but  not  necessary,  to  lay  off 
the  positive  numbers  to  the  right  of  0  and  the  negative  numbers 
to  the  left.  Then  any  point  M  on  the  scale  represents  a  real 
number,  namely,  the  number  which  measures  the  distance  of  M 
from  0 ;  positive  if  M  is  to  the  right  of  0,  and  negative  if  M  is 
to  the  left  of  0.  Conversely,  any  real  number  is  represented  by 
one  and  only  one  real  point  on  the  scale. 

11.  Zero  and  infinity.  There  are  two  mathematical  concepts 
usually  included  in  the  number  series,  for  which  special  rules  of 
operation  are  needed.  These  are  zero,  represented  by  the  symbol  0, 
and  infinity,  represented  by  the  symbol  oo. 

Zero  arises  in  the  first  place  by  subtracting  a  quantity  from  an 
equal  quantity ;  thus,  a  —  a  =  0.  It  signifies  in  this  sense  the 
absence  of  quantity  —  nothing.  It  cannot,  then,  either  operate 
upon  a  quantity  or  be  operated  upon;  for  all  operations  imply 
the  existence  of  the  quantities  concerned.     Literally,  then,  the 

n    n 

expressions  a  x  0,  -  >  -  >  are  meaningless.    However,  it  is  possible 

to  put  into  these  symbols  conventional  meanings,  as  follows: 

Take  the  three  expressions  ax,  ->  ->  and  consider  what  hap- 

a    X 

pens  when  x  is  taken  smaller  and  smaller,  constantly  nearer  to 
zero  but  never  equal  to  it.  It  requires  only  elementary  arith- 
metic to  see  that  ax  and  -  may  each  be  made  as  small  as  we 

a 


30  GRAPHICAL  KEPllESENTATlOX 

please  by  taking  a;  sufficiently  small,  while  -  becomes  indefinitely 

great  as  x  decreases,  and  may  be  made  larger  than  any  quantity 
we  may  choose  to  name.  We  may  express  the  first  two  results 
concisely  by  the  formulas 

a  X  0  =  0,         -  =  0. 
a 

We  can  express  the  last  result  in  a  formula,  however,  only  by 
introducing  the  concept  infinity.  Wlien  the  value  of  a  quantity 
is  indefinite,  but  the  quantity  is  increasing  or  decreasing  in  such 
a  way  that  its  numerical  value  is  greater  than  any  assigned  quan- 
tity, however  great,  it  is  said  to  become  infinite.  It  is  then  denoted 
by  the  symbol  oo,  called  infinity.  We  can  accordingly  express  our 
third  result  by  the  formula 

a 

0  =  "' 

which  means  that  when  the  denominator  of  a  fraction  decreases,  be- 
coming constantly  nearer  to  zero,  the  value  of  the  fraction  increases 
and  becomes  greater  than  any  quantity  which  can  be  named. 

The  symbols  a  ^ 

a  X  00,  —  >  — 

00  a 

are  also  literally  meaningless.  We  can,  however,  give  a  conven- 
tional meaning  to  them  by  writing  ax,  ->  -,  and  studying  the 

X    a 

effect  of  increasing  x  indefinitely.  Elementary  arithmetic  leads 
to  the  results  expressed  by  the  formulas 

a  X  00  =  oo,         —  =  0,  —  =  00. 

CO  a 

Two  other  forms  also  occur  in  practice,  namely,  -  and  —  •    These 

0  00 

arise  when  we  have  a  fraction  -  in  which  the  numerator  and 

the  denominator  either  approach  zero  together  or  increase  indefi- 
nitely together.  The  value  of  the  fraction  cannot  be  determined 
unless  we  know  a  law  to  govern  x  and  y.  These  fractions  are 
consequently  called  indeterminate  forms,  and  will  be  considered 
later  in  the  course. 


COMPLEX  NUMBERS  31 

Neither  zero  nor  infinity  can  be  said  to  have  an  intrinsic  alge- 
braic sign.  In  some  cases  a  quantity  may  increase  in  value, 
remaining  always  positive.  It  is  then  said  to  be  +  co.  At  other 
times  it  may  increase  numerically,  remaining  always  negative. 
It  is  then  said  to  be  —  co.  Often,  however,  the  quantity  is  iadefi- 
nitely  great  in  such  a  way  that  the  sign  is  ambiguous.  An 
example  is  tan  90°.  If  an  acute  angle  is  made  nearer  and  nearer 
to  90°,  its  tangent  increases  indefinitely,  remaining  positive.  But 
if  an  obtuse  angle  is  made  nearer  and  nearer  to  90°,  its  tan- 
gent increases  indefinitely,  remaining  negative.  Hence  we  say 
tan  90°  =  00,  and  no  algebraic  sign  can  be  attached  to  it. 

Similar  considerations  hold  for  the  sign  of  zero. 

12.  Complex  numbers.  If  one  restricts  himself  to  the  use  of 
the  real  numbers,  named  in  §  10,  it  is  impossible  to  perform  the 
operation  of  evolution  without  exception ;  for  the  even  root  of  a 
negative  number  is  not  a  real  number.  It  is  therefore  necessary, 
if  the  generality  of  all  algebraic  operations  is  to  be  maintained,  to 
introduce  a  new  kind  of  number,  called  a  complex  number.  These 
numbers  will  be  used  very  little  in  this  volume,  and  the  following 
resume  of  the  matter  usually  contained  in  algebra  is  sufficient  for 
our  present  purposes.  A  further  discussion  will  be  given  in  the 
second  volume. 

The  imaginary  unit  is  V—  1,  and  is  denoted  by  i.    Then 

e=-i. 

By  multiplying  this  equation  successively  by  i,  we  find 
i^=—  i,        i*  =  1,        i^  =  i,        i^=  —  l,         •  •  • ; 
and,  in  general,  —  — 

i''  =  l,         'i"  +  '  =  t,         i"'^'  =  -l,         i^^+'"  =  -i, 

where  k  is  zero  or  any  integer. 

If  h  is  any  real  number,  the  product  M  is  called  a  pure  imagi- 
nary number.  The  square  root  of  any  negative  number  is  pure 
imaginary ;  thus, 


32  GRAPHICAL  REPRESENTATION 

If  a  and  h  are  any  two  real  numbers,  the  combination  a  +  bi 
is  caUed  a  complex  imaginary  number,  or,  more  simply,  a  complex 
number.  A  complex  number  reduces  to  a  pure  imaginary  number 
when  a  =  0,  and  to  a  real  number  when  5  =  0.  If  a  =  0  and  5  =  0, 
the  complex  number  a  +  bi  =  0;  and  conversely,  if  a  +  bi  =  0, 
then  a  =  0  and  5  =  0. 

All  operations  with  complex  numbers  are  carried  out  by  using 
the  ordinary  laws  of  algebra  and  replacing  all  powers  of  i  by  their 
values  just  determined. 

Ex.  1.   V-3  X  V32  z=  iVs  X  i  V2  =  ?:2  Ve  =  -  Ve. 

3j-V34_3  +  2i      2  +  2i_6  +  10i  +  4i2      2  +  lOi      l+sV^ 


Ex.  2. 


4      2-2i      2  +  2i  4-4  i^ 


Two  complex  numbers  such  as  a  +  5*  and  a  —  bi,  where  a  and  b 
have  the  same  values  in  each,  are  called  conjugate  complex  numbers. 
Their  product  is  a  real  number ;  thus, 

(a-{-bi)(a-bi)  =  a^-\.b\ 

It  is  clear  that  the  complex  numbers  have  no  place  on  the  num- 
ber scale  of  §  10. 

13.  Addition  of  segments  of  a  straight  line.  Consider  any 
straight  line  connecting  two  points  A  and  B.  In  elementary 
geometry  only  the  position  and  the  length  of  the  line  are  consid- 
ered, and  consequently  it  is  immaterial  whether  the  line  be  called 
AB  or  BA  ;  but  in  the  work  to  follow  it  is  often  important  to  con- 
sider the  direction  of  the  line  as  well.  Accordingly,  if  the  direction 
of  the  line  is  considered  as  from  A  to  B,  it  is  called  ^j5;  but  if 
the  direction  is  considered  from  B  to  A,  it  is  called  BA.  It  will 
^e   seen  later  that  the  distinction 

-4  B  c         between  AB  and  BA  is  the  same 

Fig.  2  ^^  t-hat  between   +  a  and  —  a  in 

algebra. 

Consider  now  two  segments  AB  and  BC  on  the  same  straight 
line,  the  point  B  being  the  end  of  the  first  segment  and  the  begin- 
ning of  the  second.  The  segment  AC  is  called  the  sum  of  AB  and 
BC,  and  is  expressed  by  the  equation 

AB+BC=AC.  (1) 


SEGMENTS  OF  A  STEAIGHT  LINE  33 

This  is  clearly  true  if  the  points  are  in  the  position  of  fig.  2,  but 
it  is  equally  true  when  the  points  are  in  the  position  of  fig.  3, 
Here  the  line  BC,  being  opposite  in 

direction  to  AB,  cancels  part  of  it,        j[  J  ^ 

leaving  ^C.  F^,,  3 

If,  in  the  last  figure,  the  point  C 
is  moved  toward  A,  the  sum  AC  becomes  smaller,  until  finally 
when  C  coincides  with  A  we  have 

AB  +  BA  =  0,     or     BA=-AB.  (2) 

If  the  point  C  is  at  the  left  of  A,  as  in  fig.  4,  we  still  have 
4B  +  BC  =  AC,  where  AC  =  -CAhy  (2).  , 

It  is  evident  that  this  addition 
— ^  '^  ^         is  analogous  to  algebraic  addition, 

■pj^  ^  and  that  this  sum  may  be  an  arith- 

metical difference. 
From  (1)  we  may  obtain  by  transposition  a  formula  for  sub- 
traction, namely, 

BC  =  AC-AB.  (3) 

This  is  universally  true  since  (1)  is  universally  true. 

This  result  is  particularly  important  when  applied  to  segments 
of  the  number  scale  of  §  10.  For  if  x  is  any  number  corresponding 
to  the  point  M,  we  may  always  place  x  =  OM,  since  both  x  and  OM 
are  positive  when  M  is  at  the  right  of  0,  and  both  x  and  OM  are 
negative  when  M  is  at  the  left  of  0.  Now  let  31^  and  M^  be  any 
two  points,  and  let  x^  =  OM^  and  x^  =  OM^.    Then 

3I^M^  =  OM^  -  OM^  =  «2  -  ^r 
On  the  other  hand, 

M^M^  =  OM^  —  031^  =  x^—x,_=  —  M^M^. 

It  is  clear  that  the  segment  31^31^  is  positive  when  31^  is  at  the 
right  of  Jtfp  and  is  negative  when  M„  is  at  the  left  of  31^ 

Hence,  the  length  and  the  sign  of  any  segment  of  the  number 
scale  is  found  ly  subtracting  the  value  of  the  x  corresponding  to 
the  beginning  of  the  segment  from  the  value  of  the  x  corresponding 
to  the  end  of  the  ser^m^ent. 


34 


GRAPHICAL  REPKESENTATION 


14.  Projection.  Let  AB  and  MN  (figs.  5,  6)  be  any  two  straight 
lines  in  the  same  plane,  the  positive  directions  of  which  are  respec- 
tively AB  and  MN.  From  A  and  B  draw  straight  lines  perpendicu- 
lar to  3IJSf,  intersecting  it  at  points  A'  and  B'  respectively.   Then  A'  B' 


is  the  projection  oi  AB  on  MN,  and  is  positive  if  it  has  the  direction 
MN  (fig.  5),  and  is  negative  if  it  has  the  direction  NM  (fig.  6). 

Denote  the  angle  between  MN  and  AB  by  ^,  and  draw  A  C  par- 
allel to  3IN.    Then  in  both  cases,  by  trigonometry, 

AC=ABcos(f>. 

But  AC=A'B',  and  therefore 

A'B'  =  AB  cos  </). 

Hence,  to  find  the  projection  of  one  straight  line  upon  a  second, 
multiply  the  length  of  the  first  hy  the  cosine  of  the  angle  hetiveen  the 
2)ositive  .directions  of  the  two  lines. 

Ex.  It  is  customary  in  meclianics  to  represent  a  foi'ce  by  a  straight  line, 
tlie  lengtli  and  tlie  direction  of  wliich  denote  respectively  the  magnitude  and  the 
direction  of  the  force.  Then  the  component  of  the  force  in  any  direction  is  the 
projection  upon  that  direction  of  the  line  which  represents  the  force. 


—N 


In  particular,  let  Fi  and  F2,  represented  respectively  by  AB  and  AC  (figs. 
7,  8),  be  two  forces  acting  at  A  along  the  same  line,  and  let  MN  be  a  line 
which  makes  an  angle  <f>  with  AB. 


COORDINATE  AXES 


35 


The  respective  components  of  Fi  and  F2  are  represented  by  A'B'  and  A'C, 
and  the  resultant  component  is  represented  by  A'B'  +  A'C". 

But  A'B'=Fi  cos  0,  and  ^'(7'= F2  cos  (j> ;  hence,  by  substitution,  tlie  resultant 
component  is  F^  cos  0  +  jFs  cos  cj).  It  is  to  be  noted  that  in  fig.  8  Fi  and  F2  have 
opposite  signs. 

15.  The  projection  of  a  broken  line  upon  a  straight  line  is  defined 
as  the  algebraic  sum  of  the  projections  of  its  segments. 

Let  ABCDE  (fig.  9)  be  a  broken  line,  MN  o.  straight  liue  in  the 
same  plane,  and  AE  the  straight  line 
joining  the  ends  of  the  broken  line. 

Draw  AA',  BB',  CC,  BD\  and 
EE'  perpendicular  to  MN',   then  ^^_ 
A!B\  B'C,  CD',  J)'E\  and  A'E'  ^ 
are  the  respective  projections   on 
MNoi  AB,  BC,  CD,  DE,  and  AE. 


But 


A'B'  +  B'C  +  CD'  +  D'E'  =  A'E'. 


(^T  §  13) 


Hence,  the  projection  of  a  broken  line  upon  a  straight  line  is 
equal  to  the  projection  of  the  straight  line  joining  its  extremities. 

Ex.  If  ABCDE  (fig.  9)  represents  a  polygon  of  forces,  we  have  the  result: 
the  component  of  the  resultant  in  any  direction  is  the  sum  of  the  components 
of  the  forces  in  that  direction. 


X- 


M 


16.  Coordinate  axes.    Let  X'X  and  Y'  Y  be  two  number  scales 
at  right  angles  to  each  other,  with  their  zero  points  coincident  at  0, 

as  in  fig.  10. 

Let  P  be  any  point  in  the 
plane,  and  through  P  draw 
straight  Imes  perpendicular  to 
X'X  and  Y'Y  respectively, 
intersectmg  them  at  M  and  iV". 
If  now,  as  in  §  13,  we  place 
X  =  OM,  and  y  —  ON,  it  is 
clear  that  to  any  point  P  there 
corresponds  one  and  only  one 
pair  of  numbers  x  and  y,  and 
to  any  pair  of  numbers  corresponds  one  and  only  one  point  P. 


Y' 
Fig.  10 


X 


36  GRAPHICAL  REPRESENTATION 

If  a  point  P  is  given,  x  and  y  may  be  found  by  drawing  the  two 
perpendiculars  MP  and  NP  as  above,  or  by  drawing  only  one  per- 
pendicular as  MP.    Then  MP  =  OiV^=  y  and  0M=  x. 

On  the  other  hand,  if  x  and  y  are  given,  the  point  P  may  be 
located  by  finding  the  points  M  and  N  corresponding  to  the  num- 
bers X  and  y  on  the  two  number  scales,  and  drawing  perpendiculars 
to  X'X  and  Y'  Y  respectively  through  M  and  N.  These  perpen- 
diculars intersect  at  the  required  point  P.  Or,  as  is  often  more 
convenient,  a  point  M  corresponding  to  x  may  be  located  on  its 
number  scale,  and  a  perpendicular  to  X'X  may  be  drawn  through 
M,  and  on  this  perpendicular  the  value  of  y  laid  off.  In  fig.  10, 
for  example,  M  corresponding  to  x  may  be  found  on  the  scale  X'X, 
and  on  the  perpendicular  to  X'X  at  M,  MP  may  be  laid  off  equal 
to  y.  When  the  point  is  located  in  either  of  these  ways  it  is  said 
to  be  plotted.  It  is  evident  that  plotting  is  most  conveniently  per- 
formed when  the  paper  is  ruled  in  squares,  as  in  fig.  10. 

These  numbers  x  and  y  are  called  respectively  the  abscissa  and 
the  ordinate  of  the  point,  and  together  they  are  called  its  coordi- 
nates. It  is  to  be  noted  that  the  abscissa  and  the  ordinate,  as 
defined,  are  respectively  equal  to  the  distanpes  from  Y'  Y  and  X'X 
to  the  point,  the  direction  as  weU  as  the  magnitude  of  the  distances 
being  taken  into  account.  Instead  of  designating  a  point  by  writing 
x  =  a  and  y  =  —  b,itis  customary  to  write  P(a,  —  h),  the  abscissa 
always  being  written  first  in  the  parenthesis  and  separated  from 
the  ordinate  by  a  comma.  X'X  and  Y'  Y  are  called  the  axes  of 
coordinates,  but  are  often  referred  to  as  the  axes  of  x  and  y 
respectively. 

17.  Distance  between  two  points.  Let  P^{x^,  y^)  and  P,{x^,y^ 
be  two  points,  and  at  first  assume  that  P^P^  is  parallel  to  one  of 
the  coordinate  axes,  as  OX  (fig.  11).  Then  y^^y^  Now  M^M^, 
the  projection  of  P^P^  on  OX,  is  evidently  equal  to  P^P^.  But 
Y  M^M^  =  x^-x^{^  13).    Hence 

I\T^=i     X^  Xy  (1) 

In  like  manner,  if  x^  =  x^  P^P^  is  parallel 
to  OY,  and 

Fio.ii  P.n=^y-yy  (2) 


^. 

R 

Ml    0 

M,    ^ 

DISTANCE  BETWEEN  TWO  POINTS 


37 


If  x^  4--  x^  and  y^  ^  y^,  P^P^  is  not  parallel  to  either  axis.  Let 
the  points  be  situated  as  in  fig.  12,  and  through  ij  and  P^  draw 
straight  lines  parallel  respectively  to  OX  and  OY.  They  will  meet 
at  a  point  R,  the  coordinates  of  which  are  readily  seen  to  be 
{x„  y,).    By  (1)  and  (2), 

P^B  =  x^—  x^,         RP^  =  ^2  -  Vv 
But  in  the  right  triangle  P^RIl, 


PP  = 


whence,  by  substitution,  we  have 


Fig. 12 

PxP.  =  ^(^2 -*'i)'+ (2/2-2/1)'-  (3) 


It  is  to  be  noted  that  there  is  an  ambiguity  of  algebraic  sign  on 
account  of  the  radical  sign.  But  since  P^Il  is  parallel  to  neither 
coordinate  axis,  the  only  two  directions  in  the  plane  the  positive 
directions  of  which  have  been  chosen,  we  are  at  liberty  to  choose 
either  direction  of  P^P^  as  the  positive  direction,  the  other  becoming 
the  negative. 

It  is  also  to  be  noted  that  formulas  (1)  and  (2)  are  particular 
cases  of  the  more  general  formula  (3). 

Ex.    Find  the  coordinates  of  a  point  equally  distant  from  the  three  points 
Pi(l,  2),  P2(-  1,  -  2),  and  Pz(2,  -  5). 
Let  P  (x,  y)  be  the  required  point.    Then 


PiP=  P^P  and  P2P  =  P3P. 


But 


PiP  =  V(x  -  1)2  +  {y-  2)2, 


P2P  =  V(a;  +  l)2  +  (2/  +  2)2, 


P3P  =  V(x  -  2)2  +  (y+  5)2. 


V(x  -  1)2  +  {y-  2)2  =  V(x  +  1)2  +  (y  +  2)2, 


V(a;  +  1)2  +  {y  +  2f  =  V(x  -  2)2  +  (y  +  5)2, 
ehence,  by  sohition,  x  =  f  and  y  -  -  ^.     Therefore  the   required  point  is 


38 


GRAPHICAL  EEPRESENTATION 


18.  CoUinear  points.  Let  P{x,  y)  be  a  point  on  the  straight  line 
determined  hy  P^{x^,  y^)  and  Bi{x^,  y^,  so  situated  that  P^P  =  lil^P^). 

There  are  three  cases  to  consider  according  to  the  position  of 
the  point  P.    If  P  is  between  the  points  P^  and  P  (fig.  13),  the 


3 

X 

P/ 

R/ 

/        0 

M, 

M 

M,       ' 

Fig.  13 


Fig.  14 


segments  P^P  and  P^P^  have  the  same  direction,  and  P^P<P^P^; 
accordingly  Hs  a  positive  number  less  than  unity.  If  P  is  beyond 
P^  from  i^  (fig.  14),  P^P  and  P^P^  still  have  the  same  direction,  but 
P^P  >  P^P^ ;  therefore  /  is  a  positive  number  greater  than  unity. 
Finally,  if  P  is  beyond  P^  from  P^  (fig.  15), 
PyP  and  P^P,  have  opposite  directions,  and 
/  is  a  negative  number,  its  numerical  value 
ranging  all  the  way  from  0  to  oc. 

In  the  first  case  P  is  called  a  point  of 
internal  division,  and  in  the  last  two  cases 
it  is  called  a  point  of  external  division. 

In  all  three  figures  draw  P^M^,  PM, 
and  I^M^  perpendicular  to  OX.  In  each 
figure  OM=OM^  +  M^M;  and  since  P^P  =  I  (P^P,),  M^M=  1{M^M^, 
by  geometry. 

.  • .  OM  =  OM^  +  /  {M^M^), 


I 


Fig.  15 


whence,  by  substitution, 


x  =  x^-\-  l(x2—x^). 


(1) 


By  drawing  lines  perpendicular  to  OY  we  can  prove,  in  the 
same  way, 

y  =  y.+  i{y.-yx)-  (2) 


COLLINEAE  POINTS 


39 


In  particular,  if  P  bisects  the  line  P^P^,  I  =  ^,  and  these  formulas 
become 

2-^2 


X  = 


Ex.  1.  Find  the  coordinates  of  a  point  J  of  the  distance  from  Pi  (2,  3)  to 
P2(3,  -3). 

If  the  required  point  is  P(x,  y), 

X  =  2  +  f  (8  -  2)  =  22, 
2/  =  3  +  -?(-3-3)  =  f. 


Ex.  2.    Prove  analytically  that  the  straight  line  dividing  two  sides  of  a  tri- 
angle in  the  same  ratio  is  parallel  to  the  third  side. 

Let  one  side  of  the  triangle  coincide  with  OX,  one  vertex  being  at  O.  Then 
the  vertices  of  the  triangle  are  0(0,  0),  A{xi,  0), 
B{X2,  2/2)   (fig.  16).    Let  CD  divide  the  sides  OB 
and  AB  so  that  OC  =  1{0B)  and  AD  =  1{AB). 

If  the  coordinates  of  C  are  denoted  by  (X3,  1/3) 
and  those  of  D  by  (X4,  2/4),  then,  by  the  above 
formulas, 

Xa  =  1x2,  2/3  =  ly2, 

and  X4  =  Xi  +  Z(xo  —  xi),  2/4  =  l)/2- 

Since  7/3  =  2/4,  CD  is  parallel  to  OA.  Fig.  16 


19.  Let  us  now  see  what  happens  as  different  real  values  are 
assigned  to  I.    Wlien  I  =  0,  P  coincides  with  P^  (fig.  17).    As  / 

increases    in    value,    the 
point  P  moves  along  the 
line  toward  P^  till,  when 
/  =  1,  it  coincides  with  P,. 
As  the  value   of  I  con- 
tinues   to    increase,    the 
point  P  continues  to  move 
along  the  line  away  from 
^  and  in  the  same  direc- 
tion as  before. 
If  negative  values  are  assigned  to  /,  in  ascending  order  of  numer- 
ical magnitude,  the  point  P  moves  along  the  line,  away  from  P^,  in 
the  opposite  direction  from  P,. 


Fig.  17 


40  GRAPHICAL  REPRESENTATION 

It  follows  that 

^1  +  ^ («2 -  ^i)     and     y^  +l{y^—  2/1) 

may  be  made  to  represent  the  coordinates  of  any  point  of  the 
straight  line  determmed  by  the  pomts  P^  and  i^  by  assigning  the 
appropriate  value  to  I,  the  range  of  values  for  each  segment  of 
the  line  being  indicated  in  fig.  17. 

Ex.  Consider  the  straight  line  determined  by  the  two  points  Pi(—  1,  —  4) 
and  P2(5,  6).    Any  other  point  P  on  tliis  line  has  the  coordinates 

x  =  -l  +  6Z,  ^  =  -44-10^. 

When  Z  <  0,  it  is  clear  that  a;<  —  1,  y  <  —  4k;  hence  P  lies  at  the  left 
of  Pi.  When  0  <  i  <  1,  it  is  clear  that  — l<x<5,  — 4<2/<6;  hence  P 
lies  between  Pi  and  P2.  When  l>\,  it  is  clear  that  a;  >  5,  y  >Q;  hence 
P  lies  at  the  right  of  P2. 

20.  Variable  and  function.  A  quantity  which  remains  un- 
changed throughout  a  given  problem  or  discussion  is  called  a 
constant.  A  quantity  which  changes  its  value  in  the  course  of 
a  problem  or  discussion  is  called  a  variable.  If  two  quantities 
are  so  related  that  when  the  value  of  one 'is  given  the  value  of 
the  other  is  determined,  the  second  quantity  is  called  a  function 
of  the  first.  Wlien  the  two  quantities  are  variables  the  first  is 
called  the  independent  variable,  and  the  function  is  sometimes 
called  the  dependent  variable.  As  a  matter  of  fact,  when  two 
related  quantities  occur  in  a  problem  it  is  usually  a  matter  of 
choice  which  is  called  the  independent  variable  and  which  the 
function.  Thus,  the  area  of  a  circle  and  its  radius  are  two 
related  quantities  such  that  if  one  is  given  the  other  is  deter- 
mined. We  can  say  that  the  area  is  a  function  of  the  radius, 
and  likewise  that  the  radius  is  a  fvmction  of  the  area. 

The  relation  between  the  independent  variable  and  the  function 
can  be  graphically  represented  by  the  use  of  rectangular  coordi- 
nates. For,  if  we  represent  the  independent  variable  by  x  and  the 
corresponding  value  of  the  function  by  y,  x  and  y  will  determine 
a  point  in  the  plane,  and  a  number  of  such  points  will  outline  a 
curve  indicating  the  correspondence  of  values  of  variable  and 
function.    This  curve  is  called  the  graph  of  the  function. 


EXAMPLES  OF  FUNCTIONS 


41 


Ex.  1,  An  important  use  of  the  graph  of  a  function  is  in  statistical  work. 
The  following  table  shows  the  price  of  standard  steel  rails  per  ton  in  the 
respective  years: 


1895 $24.33 

1896 28.00 

1897 18.75 

1898 17.62 

1899 28.12 


1900 $32.29 

1901 27.33 

1902 28.00 

1903 28.00 

1904 28.00 


If  we  plot  the  years  as  abscissas,  calling  1895  the  first  year,  1896  the  second 
year,  etc.,  and  plot  the  price  of  rails  as  ordinates,  making  one  unit  of  ordinates 
correspond  to  ten  doUare,  we  shall  locate  the  points  Pi,  P^, . . .,  Pio  in  fig.  18.  In 
order  to  study  the  variation  in  price,  we  join  these  points  in  succession  by  straight 


Fig.  18 


lines.  The  resulting  broken  line  serves  merely  to  guide  the  eye  from  point  to 
point,  and  no  jwint  of  it  except  the  vertices  has  any  other  meaning.  It  is  to 
be  noted  that  there  is  no  law  connecting  the  price  of  rails  with  the  year. 
Also  the  nature  of  the  function  is  such  that  it  is  defined  only  for  isolated 
values  of  x. 

Ex.  2.    As  a  second  example  we  take  the  law  that  the  postage  on  each  ounce  or 
fraction  of  an  ounce  of  fii-st-class  mail  matter  is  two  cents.    The  postage  is  then  a 
known  function  of  the  weight.    Denoting  each 
ounce  of  weight  by  one  unit  of  x,  and  each  two        y 
cents  of  postage  by  one  unit  of  y,  we  have  the 
series  of  straight  lines  (fig.  19)  parallel  to  the 
axis  of  X,  representing  corresponding  values  of 
weight  and  postage.    Here  the  function  is  defined 
by  United  States  law  for  all  positive  values  of  x, 
but  it  cannot  be  expressed  in  elementary  mathe- 
matical .symbols.    A  peculiarity  of  the  graph  is 
the  series  of  breaks.    The  lines  are  not  connected , 
but  all  points  of  each  line  represent  correspond- 
ing values  of  x  and  y.  Fig.  19 


O 


42 


GKAPHICAL  KEPRESENTATION 


Ex.  3.  As  a.  third  example,  differing  in  type  from  eacli  of  the  preceding,  let 
us  take  the  following.  While  it  is  known  that  there  i.s  some  physical  law  con- 
necting the  pressure  of  saturated  steam  with  its  temperature,  so  that  to  every 
temperature  there  is  somo  corresponding  pressure,  this  law  has  not  yet  been 
formulated  mathematically.  ^'everthele.ss,  knowing  some  corresponding  values 
of  temperature  and  prassure,  we  can  construct 
a  curve  that  is  of  considerable  value.  In  the 
table*  below,  the  temperatures  are  in  degrees 
Centigrade  and  the  pressures  are  in  millimeters 
of  mercury. 


Y 

J 

7 

r 

j 

7 

t 

zr 

1 

t 

^ 

i 

.1 

r 

1 

J 

r 

r 

t 

t. 

1     _ 

t      - 

J 

t          _ 

-    4 

J. 

J 

r 

ol 



1       1    X 

Temperature 

Pressub 

100 

760 

105 

906 

110 

1074.7 

115 

1268.7 

120 

1490.5 

125 

1743.3 

130 

2029.8 

135 

2353.7 

140 

2717.9 

146 

3126.1 

150 

3581.9 

Let  100  represent  the  zero  point  of  tempera- 
ture, and  let  each  unit  of  x  represent  5  degrees 
of  temperature ;  also  let  each  unit  of  y  represent 
100  millimeters  of  pressure  of  mercury,  and  locate 
the  points  representing  the  corresponding  values 
of  temperature  and  pre.ssure  given  in  the  above 
table.  Through  the  points  thus  located  draw  a 
smooth  curve  (fig.  20)  i.e.  one  which  has  no  sudden 
changes  of  direction.  AVhile  only  the  eleven  points 
located  are  exact,  all  other  points  are  approxi- 
mately accurate,  and  the  curve  may  be  used  for 
approximate  computation  as  follows :  A-ssume  any 
temperature,  and,  laying  it  oif  as  an  abscissa, 
measure  the  corresponding  ordinate  of  the  curve. 
While  not  exact,  it  will,  nevertheless,  give  an  approximate  value  of  the  corre- 
sponding pressure.  Similarly,  a  pre.ssure  may  be  assumed  and  the  corresponding 
temperature  determined.  It  may  be  added  that  the  more  closely  together  the 
tabulated  values  are  taken,  the  better  the  approximation  from  the  curve,  but 
the  curve  can  never  be  exact  at  all  points. 


Fig.  20 


»From  C.  H.  Peabody's  "  Steam  Tables,"  computed  for  sea  level  at  a  latitude  of 
45  degrees. 


CLASSES  OF  FUNCTIONS 


43 


Ex.  4.  As  a  final  example,  we  will  take  the  law  of  Boyle  and  Mariotte  for  per- 
fect gases,  namely,  at  a  constant  temperature  the  volume  of  a  definite  quantity 
of  gas  is  inversely  proportional  to  its 
pressure.  It  follows  that  if  we  repre- 
sent the  pressure  by  x  and  the  corre- 

k 
spending  volume  by  y.  then  y  =  -, 

X 

where  fc  is  a  constant  and  x  and  y  are 
positive  variables.  A  curve  (fig.  21)  in 
the  first  quadrant,  the  coordinates 
of  every  point  of  which  satisfy  this 
equation,  represents  the  comparative 
changes  in  pressure  and  volume,  show- 
ing that  as  the  pressure  increases  by  a 
certain  amount  the  volume  is  decreased 
more  or  less,  according  to  the  amount 
of  pressure  previously  exerted. 

This  example  differs  from  the  pre- 
ceding in  that  the  law  of  the  function 

is  fully  known  and  can  be  expressed  in  a  mathematical  formula.  Consequently, 
we  may  find  as  many  points  on  the  curve  as  we  please,  and  may  therefore  con- 
struct the  curve  to  any  required  degree  of  accuracy. 


Fig.  21 


21.  Classes  of  functions.  We  shall  consider  in  this  book  only 
those  functions  of  one  variable  which  can  be  expressed  by  means 
of  elementary  mathematical  symbols.  The  simplest  kind  of  such 
functions  is  the  algebraic  polyTiomial,  expressed  by 

ao«"-f  ftj^t-"-^  +  •  •  •  +  «„_!«  +  «„, 

where  all  the  exponents  are  positive  integers  and  the  coefficients 
a^,  «!,  •••,«„_ J,  «„  are  real  or  complex  numbers  or  zero.  The 
number  n  is  the  degree  of  the  polynomial.  These  functions  are 
discussed  in  Chaps.  Ill  and  IV. 

The  quotient  of  two  algebraic  polynomials  is  a  rational  algebraic 
fraction,  expressed  by 

«„»"  -f  ajiz;"""^  +•••  +  «„_]''»  +  «„ 


Examples  of  functions  of  this  kind  are  discussed  in  Chap.  VL 


44  GKAPHICAL  REPRESENTATION 

If  a  function  requires  for  its  expression  the  use  of  radical  signs 
combined  with  algebraic  polynomials,  it  is  an  example  of  an  irra- 
tional algebraic  function ;  for  example. 


Ab  +  JIZI. 

Examples  of  such  functions  are  found  in  Chap.  VI. 

The  general  definition  of  an  algebraic  function  is  given  in 
Chap.  IX,  and  examples  of  non-algebraic,  or  transcendental  func- 
tions, are  given  in  Chap.  XIII. 

22.  Functional  notation.  When  y  is  a  function  of  x  it  is  cus- 
tomary to  express  this  by  the  notation 

Then  the  particular  value  of  the  function  obtained  by  giving  x  a 
definite  value  a  is  written  f{a).    For  example,  if 

tben  /(2)  =  2«+3-22-f-l=21, 

/(0)=0«+3-0^+l  =  l, 
/(-3)  =  (-3)H3(-3)^+l  =  l, 
/(a)  =  a'+3a2-f  1. 

If  more  than  one  function  occurs  in  a  problem,  one  may  be 
expressed  asf(x),  another  as  F{x),  another  as  <l>(x),  and  so  on.  It 
is  also  often  convenient  in  practice  to  represent  different  functions 
by  the  symbols  f(x),  f^(x),  f^{x),  etc. 

If /(a;)  is  any  function,  and  we  place 

y  =/('«), 

we  may,  a^  already  noted,  construct  a  curve  which  is  the  graph  of 
the  function.  The  relation  between  this  curve  and  the  equation 
y  =f{x)  is  such  that  all  points  the  coordinates  of  which  satisfy  the 
equation  lie  on  the  curve ;  and  conversely,  if  a  point  lies  on  the 
curve,  its  coordinates  satisfy  the  equation. 


PROBLEMS  45 

The  curve  is  said  to  be  represented  by  the  equation,  and  the  equa- 
tion is  called  the  equation  of  the  curve.  The  curve  is  also  called 
the  locus  of  the  equation.  Its  use  is  twofold,  —  on  the  one  hand, 
we  may  study  a  function  by  means  of  the  appearance  and  the 
properties  of  the  curve,  and,  on  the  other  hand,  we  may  study  the 
geometric  properties  of  a  curve  by  means  of  its  equation.  Both 
methods  will  be  illustrated  in  the  following  pages. 

PROBLEMS 

1.  Find  the  perimeter  of  the  triangle  the  vertices  of  which  are  (2,  3), 
(-  3,  3),  (1,  1). 

2.  Prove  that  tlie  triangle  the  vertices  of  which  are  (—  4,   —  3),  (2,  1), 
( —  5,  5)  is  isosceles. , 

3.  Prove  that  (6,  2),  (-  2,  -  4),  (5,  -  5),  (-1,  3)  are  points  of  a  circle  the 
center  of  which  is  (2,  —  1).    What  is  its  radius  ? 

4.  Prove  that  the  quadrilateral  of  which  the  vertices  are  (2,  2),  (4,  5), 
(—1,  4),  (—  3,  1)  is  a  parallelogram. 

5.  Find  a  point  equidistant  from  the  points  (—3,  4),  (5,  3),  and  (2,  0). 

6.  Find  the  center  of  a  circle  passing  through  the  points  (0,  0),  (—3,  3), 
and  (5,  4). 

7.  Find  a  point  on  the  axis  of  x  which  is  equidistant  from  (0,  4)  and 
(-3,-3). 

8.  A  point  is  equally  distant  from  the  points  (1,  1)  and  (—  2,  3),  and  its 
distance  from  OY  is  twice  its  distance  from  OX.    Find  its  coordinates. 

9.  Find  the  points  which  are  4  units  distant  from  (2,  3)  and  5  units  distant 
from  the  axis  of  y. 

10.  A  point  of  the  straight  line  joining  the  points  (—4,  —  2)  and  (4,  —  6) 
divides  it  into  segments  which  are  in  the  ratio  3  :  6.    What  are  its  coordinates  ? 

11.  Find  the  coordinates  of  a  point  P  on  the  straight  line  determined  by 
Pi (2,  -  1)  and  P2 (-  4,  5),  when  |^  =  ^ • 

12.  On  the  straight  line  determined  by  the  points  Pi (2,  4)  and  P2(  — 1,  —  3) 
find  the  point  three  fourths  of  the  distance  from  Pi  to  P2. 

13.  If  P  (x,  y)  is  a  point  on  the  straight  line  determined  by  Pi  (xi,  2/1)  and 
■P2(X2,  2/2),  such  that  — ^  =  -,  prove 

PP2         ^2 

hxi  +  hxi  ?i2/2  +  hVi 

x  = 1  w  = • 

h  +  I2  k  +  k 


46  GKAPHICAL  REPRESEKTATION 

14.  The  middle  point  of  a  certain  line  is  (1,  2)  and  one  end  is  the  point 
(—3,  5).    Find  the  coordinates  of  the  other  end. 

15.  To  what  point  must  the  line  drawn  from  (1,  —1)  to  (—4,  5)  be  extended 
in  the  same  direction  that  its  length  may  be  trebled  ? 

16.  One  end  of  a  line  is  at  (2,  —  6)  and  a  point  one  fourth  of  the  distance 
to  the  other  end  is  (—  1,  4).    Find  the  coordinates  of  the  other  end  of  the  line. 

17.  Find  the  points  of  trisection  of  the  line  joining  Pi(0,  3)  and  P2(6,  —  3). 

18.  Find  the  lengths  of  the  medians  of  the  triangle  (2,  1),  (0,  —3),  (—  4,  0). 

19.  Given  the  three  points  A{-  3,  3),  B{S,  1),  and  C{6,  0)  upon  a  straight 

A.D  AB 

line.    Find  a  fourth  point  D  such  that  = 

DC  BC 

20.  Given  four  points  Pj,  P2,  P3,  P4.  Find  the  point  halfway  between  Pi 
and  P2,  then  the  point  one  third  of  the  distance  from  this  point  to  P3,  and 
finally  the  point  one  fourth  of  the  distance  from  this  point  to  P4.  Show  that 
the  order  in  which  the  points  are  taken  does  not  affect  the  result. 

21.  Prove  analytically  that  if  in  any  triangle  a  median  is  drawn  from  the 
vertex  to  the  base,  the  sum  of  the  squares  of  the  other  two  sides  is  equal  to 
twice  the  square  of  half  the  base  plus  twice  the  square  of  the  median. 

22.  Prove  analytically  that  the  straight  line  drawn  between  two  sides  of  a 
triangle  so  as  to  cut  off  the  same  proportional  parts  measured  from  their  com- 
mon vertex  is  the  same  proportional  part  of  the  third  side. 

23.  Prove  analytically  that  if  two  medians  of  a  triangle  are  equal  the  tri- 
angle is  iso.sceles. 

24.  Prove  analytically  that  in  any  right  triangle  the  straight  line  drawn 
from  the  vertex  to  the  middle  point  of  the  hypotenuse  is  equal  to  one  half  the 
hjrpotenuse. 

25.  Prove  analytically  that  the  lines  joining  the  middle  points  of  the  opposite 
sides  of  a  quadrilateral  bisect  each  other. 

26.  Show  that  the  sum  of  the  squares  on  the  four  sides  of  any  quadrilateral 
is  equal  to  the  sum  of  the  squares  on  the  diagonals,  together  with  four  times  the 
square  on  the  lin6  joining  the  middle  points  of  the  diagonals. 

27.  Prove  analytically  that  the  diagonals  of  a  parallelogram  bisect  each 
other. 

28.  Prove  analytically  that  the  line  joining  the  middle  points  of  the  non- 
parallel  sides  of  a  trapezoid  is  one  half  the  sum  of  the  parallel  sides. 

29.  OABC  is  a  trapezoid  of  which  the  parallel  sides  OA  and  CB  are  per- 
pendicular to  OC.  D  is  the  middle  point  of  AB.  Prove  analytically  that 
OD  =  CD. 


PROBLEMS 


47 


30.  The  following  table  gives  the  price  of  a  bushel  of  wheat  in  the  New 
York  market  from  1890  to  1904.     Construct  the  graph. 


1890 

.983 

1895 

.669 

1900 

.804 

1891 

1.094 

1896 

.781 

1901 

.803 

1892 

.908 

1897 

.954 

1902 

.836 

1893 

.739 

1898 

.952 

1903 

.863 

1894 

.011 

1899 

.794 

1904 

1.107 

31.  The  following  table  shows  hourly  barometric  readings  at  a  United  States 
weather  bureau  station.    Construct  the  graph. 


1  A.M. 

28.85 

9  A.M. 

29.04 

6  P.M. 

29.13 

2 

28.87 

10 

29.05 

6 

29.18 

3 

28.90 

11 

29.05 

7 

29.21 

4 

28.92 

12  m 

29.05 

8 

29.24 

5 

28.94 

1  P.M. 

29.05 

9 

29.25 

6 

28.97 

2 

29.06 

10 

29.29 

7 

28.98 

3 

29.08 

11 

29.29 

8 

29.02 

4 

29.10 

12 

29.29 

32.  The  following  table  shows  the  number  of  inches  of  rainfall  in  Boston 
during  the  years  1880-1891.    Construct  the  graph. 


1880 

38.89 

1886 

46.47 

1881 

49.22 

1887 

41.91 

1882 

48.42 

1888 

60.27 

1883 

35.56 

1889 

54.79 

1884 

53.86 

1890 

50.21 

1885 

44.07 

1891 

49.63 

33.  The  following  is  a  portion  of  a  railway  time-table.  The  letters  indicate 
stations,  and  the  adjacent  number  gives  the  distance  fi'om  A  to  each  of  the  othei' 
stations.  The  second  and  the  third  columns  give  the  times  at  which  two  trains 
running  in  opposite  directions  leave  each  of  the  stations.  Make  a  graph  showing 
the  motion  of  each  train  and  thus  determine  the  time  and  place  of  their  passing. 


A 

10.45  AM. 

2.00 

F99 

1.06  P.M. 

10.48 

B  21 

1.30 

G  126 

9.53 

C44 

11.50 

12.56 

H  151 

2.59 

8.56 

D64 

12.11  P.M. 

1177 

7.48 

E  84 

11.30  A.M. 

K200 

4.15 

7.00  A.M. 

48 


GRAPHICAL  REPRESE:N^TATI0X 


34.  The  following  table  shows  the  amount  of  SI. 00  put  at  interest  at  4% 
compounded  annually.    Construct  the  graph. 


6yr. 

1.217 

30  yr. 

3.242 

10 

1.480 

35 

3.946 

16 

1.801 

40 

4.801 

20 

2.191 

45 

5.841 

25 

2.666 

50 

7.116 

35.  Make  a  graph  showing  the  relation  between  the  side  and  the  area  of  a 
square. 

36.  Make  a  graph  showing  the  relation  between  the  radius  and  the  area 
of  a  circle. 

37.  Make  a  graph  showing  the  relation  between  the  radius  and  the  volume 
of  a  sphere. 

38.  The  space  s  through  which  a  body  falls  from  rest  in  t  seconds  is  given 
by  the  formula  s  =  I  gt^.    Assuming  g  =  32,  construct  the  graph. 

39.  The  velocity  acquired  by  a  body  thrown  towards  the  earth's  surface 
with  a  velocity  Vq  is  given  at  the  end  of  t  seconds  by  the  formula  v  =  Vo  +  gt. 
Construct  the  graph. 

40.  Two  particles  of  mass  mi  and  mz  at  a  distance  d  from  eacK  other  attract 
each  other  with  a  force  F,  given  by  the  formula 


F  = 


cP 


Assuming  mi  =  5  and  m^  =  20,  construct  the  graph  of  F. 

41.  Ohm's  law  for  an  electric  current  is 

^  ^      Electromotive  force 

Current  = .. 

Resistance 

Assuming  the  electromotive  force  to  be  constant,  plot  the  curve  showing  the 
relation  between  the  resistance  and  the  current. 

42.  n/(x)  =  X*  -  3x2  +  7  X  -  1,  find/(3),  /(O),  f{a),  f{a  +  h). 

43.  If /(x)  =  x8  +  1,  show  that/(2)  -  4/(1)  =/(0). 

44.  If /(x)  =  x<  +  2x2  +  3,  prove  that/(-  x)  =/(x). 

45.  If /(x)  =  x6  +  3x8  -  7x,  prove  that/(-'x)  =  -/(x). 

46.  If /(x)  =  x2  -  o2,  prove  that/(a)  =/(-  a). 

4  7.  If  /i  (x)  =  x2  +  o2,  and  /2  (x)  =  2  x,  prove  that  /i  (a)  -  af.  (a)  =  0. 

48.  If /(x)  =  (x  -  I)  (x2  -  1)  (.3  _  i^^  ,  prove  that  /  (a)  =  -/(I)  . 


PROBLEMS  49 

49.  If /(x)  =  ^-^,  prove  that /(a)  •  /(-  a)  =  1. 

50.  If  fix)  =  ^'  +  2a:8  +  2x  +  l^  p^^^^  ^j^^^  ^  Jl\  ^ 

X2  \X/ 

51.  If/(x)=.^p±l,find/(3),/(0),/(-3),/(a),/Q). 

52.  If /(x)  =  |x,  prove  that  (x  +  l)/(x)  =/(x  +  1). 

53.  If /i(x)  =-v/-  +  \/|'  and  f^iz)  =  \/-  -  \/^'  P^O'^'^  ^^^^^^ 

[/i(a=)?- [/2(a;)P  =[/!(«)?• 

54.  If /(x)  =  ^^ ,  prove  that  /[/(x)]  =  x. 


CHAPTEE  III 
THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

23.  Graphical  representation.  An  algebraic  polynomial  of  the 
first  degree  is  of  the  form  mx  + 1,  where  m  and  h  are  numbers, 
which  may  be  positive  or  negative,  integral  or  fractional,  rational 
or  irrational.  We  shall  restrict  the  values  of  m  and  h,  however,  to 
real  numbers.  In  particular  cases  h  may  be  zero,  when  the  poly- 
nomial becomes  the  monomial  mx. 

To  obtain  the  graph  of  the  polynomial,  we  write 


y  =  mx  +  h, 


(1) 


and  proceed  as  in  the  examples  of  the  previous  chapter.    We  assign 
to  X  any  number  of  values  assumed  at  pleasure,  say  x^,  x^,  x^,  x^, 

etc. ;  compute  the 
corresponding  values 
of  y,  namely. 


y^  =  mx^  +  h, 
y^  =  m,x^+h, 
y^  =  mx^+h, 
y^  =  mx^+h, 


(2) 


and  plot  the  points 

m^z>    2/3).    P^i'    Vi) 

(fig.  22).     We   then 
Fig.  22  draw     the     straight 

lines  p,p„  nn,  P,P„ 

each  connecting  two  successive  points,  and  shall  prove  that  these 
lines  form  one  and  the  same  straight  line.    For  that  purpose  draw 

no 


THE  STllAIGHT  LI:NE  51 

through  each  point  Imes  parallel  to  the  coordinate  axes,  forming 
the  triangles  shown  in  fig.  22.    Then,  by  §  13, 

T[li.^  =  x,^     x^,      J^A^  =  x^      x^,      J^A^  =  x^     x^, 

^2^2  =  ^2  -  Vv  ^^Z^l  =  ^3  -  2/2 .  ^-^4^  =  2/4-2/3- 

By  subtracting  each  equation  in  (2)  from  the  one  below  it,  we 
have 

2/3-2/2  =  ^'K-«3-^"2)' 
y4-2/3  =  ^^('''4-'^3). 

Whence  |Z|l  ^  |^  ^  |Z|  =  ,,.  (4) 


or,  by  (3), 


jpi,      liR,      IIR^ 


Hence  the  triangles  of  the  figure  are  similar,  and  the  angles 
R^P^P^,  R,P,Ps,  RiPPi  are  equal.  Therefore  the  line  P^P^P^P^  is  a 
straight  line. 

Again,  let  us  take  on  this  line  any  other  point,  such  as  ^, 
which  has  not  been  used  in  constructing  the  graph,  and  draw 
JIR^  and  J?^^  parallel  to  OX  and  OY  respectively.  Then,  since 
the  triangles  PJi^P^  and  I^R^I^  are  similar, 

R,P,  _R.^J^, 

p,r,~i(k' 

that  is,  '.l^^  =.  '!^~^  =  m.  (by  (4)) 

0  4  2  1 

Therefore  y^  =  mx^—  mx^  +  y^, 

whence,  by  substituting  the  value  of  y^  given  in  (2), 

2/5  =  '^^5  +  ^' 

Hence  the  coordinates  of  P-,  satisfy  the  equation  (1). 

We  have  now  shown  that  all  points  the  coordinates  of  which 
satisfy  eqiiation  (1)  lie  on  a  straight  line,  and  that  any  point  on 


52        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

the  line  has  coordinates  which  satisfy  (1).  We  have  accordingly 
proved  the  foUowijig  proposition :  The  equation  y  =  mx  +  h  always 
represents  a  straight  line. 

24.  The  general  equation  of  the  first  degree.    The  equation 

Ax  +  By  +  C={), 

where  A,  B,  and  C  may  be  any  numbers  or  zero,  except  that 
A  and  B  cannot  be  zero  at  the  same  time,  is  called  the  general 
equation  of  the  first  degree.  We  shall  prove :  The  general  equa- 
tion of  the  first  degree  vnth  real  coefficients  always  represents  a 
straight  line. 

1.  Suppose  A  ^  {^  and  B  ^  0.  If  any  value  of  x  is  assumed, 
the  value  of  y  is  determined.  Therefore  y  is  a,  function  of  x, 
which  may  be  expressed  by  solving  the  equation  for  y ;  thus, 

A  C 

y  = X 

^  B         B 

This  equation  is  of  the  form  y  =  mx  +  h,  and  therefore  repre- 
sents a  straight  line  by  §23. 

2.  Suppose  yl  =  0,  5  T^  0.    The  equation  is  then 

By  +  C=0,     or     2/=- 1' 

B 

All  points  the  coordinates  of  which  satisfy  this  equation  lie 

on  a  straight  line  parallel  to  OX  at  a  distance units  from  it ; 

B 
and,  conversely,  any  point  on  tliis  line  has   coordinates  which 
satisfy  the  equation.    Hence  the  equation  represents  this  line. 

3.  Suppose  A^O,B  =  0.    The  equation  is  then 

Ax+C=0,     or     x  =  --, 

A 

and  represents  a  straight  line  parallel  to  0  F  at  a  distance  —  — 
units  from  it.  -^ 

Therefore  the  equation  Ax  +  By  +  C=0  always  represents  a 
straight  line. 


THE  STRAIGHT  LINE 


53 


25.  In  order  to  plot  a  straight  line  it  is,  in  general,  convenient 
to  find  the  points  L  and  K  (fig.  23),  in  which  it  cuts  OX  and  OY 
respectively.  If  the  coordinates  of  L  are  (a,  0)  and  those  of  K  are 
(0,  5),  these  coordinates  will  satisfy  the  equation  Ax  +  By+  C=  0. 
By  substitution  we  find  F 


C 

a  = > 

A 


B 


Fig.  23 


The  quantities  a  and  h,  which 
are  equal  in  magnitude  and 
sign  to  OL  and  OK  respectively, 
are  called  the  intercepts  of  the 
straight  line.  It  is  evident  that 
the  h  found  here  is  the  same  as 
in  y  =  tnx  +  h. 

If  C=0,  i.e.  if  the  equation  is  Ax-\-By=Q,  then  a  =  0  and 
5  =  0,  and  the  straight  line  passes  through  the  origin.  To  plot 
the  line,  we  must  find  by  trial  the  coordinates  of  another  point 
which  satisfy  the  equation,  plot  this  point,  and  draw  a  straight 
line  through  it  and  the  origin. 

Ex.  1.  Plot  the  line  3a;  —  52/  +  12  =  0.  Placing  y  =  0,  we  find  a  =  —  4. 
Placing  X  =  0,  we  find  b  =  2f .  We  lay  off  Oi  =  -  4,  OK  -  2f ,  and  draw  a 
straight  line  through  L  and  K. 

Ex.  2.  Plot  the  line  3x  —  5 i/  =  0.  Here  a  =  0  and  6=0.  If  we  place  x  =  1, 
we  find  2/  =  4.    The  line  is  drawn  through  (0,  0)  and  (1,  ^). 

26.  Any  straight  line  may  he  represented  hy  an  equation  of  the 
first  degree. 

The  proof  consists  in  showing  that  the  coefficients  A,  B,  C, 
in  the  general  equation  of  the  first  degree,  may  be  so  chosen  that 
the  equation  may  represent  any  straight  line  given  in  advance. 
Let  (x^,  y^)  and  (x^,  y^)  be  any  two  points  on  a  given  straight 
line.    The  coordinates  of  these  points  will  satisfy 

Ax-\-By  +  C=0,  (1) 

provided  A,  B,  C  have  such  values  that 

Ax,  +  By,+  C  =  0, 

Ax^  +  By^+C=0, 


54        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

Solving  these  equations  for  the  ratios  oiA,B,  C,  we  have  (by  §  8) 


A:B:C  = 


|y,  1 

X,     1  . 

^'i  yi 

\y.  1 

X,      1 

^2  2/2 

(2) 


If  these  values  are  used  in  (1),  that  equation  represents  a 
straight  line  which  has  two  points  in  common  with  the  given 
lin^,  and  therefore  coincides  with  it  throughout.  Hence  the 
theorem  is  proved. 

The  result  of  substituting  from  (2)  in  (1)  is 


=  0, 


which  is  the  equation  of  a  line  through  two  given  points. 

27.  Slope.  Let  P^{x^,  y^)  and  P^{x^,  y^)  (figs.  24,  25)  be  two 
points  upon  a  straight  line.  If  we  imagine  that  a  point  moves 
along  the  line  from  ij  to  i^,  the  change  in  x  caused  by  this 
motion  is  measured  in  magnitude  and  sign  by  x^—x^,  and  the 


X     y 

1 

^1     Vx 

I 

«2       ^2 

1 

Fig.  24 


Fig.  25 


change  in  y  is  measured  by  y^—y^  We  define  the  slope  of  the 
straight  line  as  the  ratio  of  the  change  in  y  to  the  change  in  x  as 
a  point  moves  along  the  line,  and  shall  denote  it  by  the  letter  m. 
"We  have  then,  by  definition, 


m  = 


2^2-^1 


ANGLES  55 

It  appears  from  equations  (4)  (§23)  that  the  letter  m  in  the 
equation  y  =  mx  +  J  has  the  meaning  just  defined.  It  follows  that 
if  the  equation  of  a  straight  line  is  in  the  form  Ax+By  +  C  =  0, 
its  slope  may  he  foimd  by  solving  the  equation  for  y  and  taking  the 
coefficient  of  x,  thus, 

AC  A 

y  = X  —  —•>     whence     m  = 

"^  B  B  B 

A  geometric  interpretation  of  the  slope  is  readily  given.  For  if 
we  draw  through  I^  a  line  parallel  to  OX,  and  through  ^  a  line 
parallel  to  0  Y,  and  call  E  the  point  in  which  these  two  lines  inter- 

BB 

sect,  then  x^—  x^  =I\B,  and  y^  —  y^  —BB^ ;  and  hence  m  =  — ^  • 

It  is  clear  from  the  figures,  as  well  as  from  equations  (4)  (§  23), 
that  the  value  of  m  is  independent  of  the  two  points  ij  and  B^ 
and  depends  only  on  the  given  line.  We  may  therefore  choose  B^ 
and  B^  (as  in  figs.  24  and  25)  so  that  I^B  is  positive.  There  are 
then  two  essentially  different  cases,  according  as  the  line  runs  up 
or  down  toward  the  right  hand.  In  the  former  case  BB^  and  m 
are  positive  (fig,  24) ;  in  the  latter  case  BJF^  and  m  are  negative 
(fig.  25).    We  may  state  this  as  foUows : 

The  slope  of  a  straight  line  is  positive  when  an  increase  in  x 
causes  an  increase  in  y,  and  is  negative  vjhen  an  increase  in  x 
causes  a  decrease  in  y. 

When  the  line  is  parallel  to  OX,  y„  =  y^,  and  consequently  m  =  0, 
as  explained  in  §  11.  If  the  line  is  parallel  to  OY,  x^=.  x^,  and 
therefore  m  =  oo  in  the  sense  of  §  11. 

28.  Angles.  The  slope  of  a  straight  line  enables  us  to  solve 
many  problems  relating  to  angles,  some  of  which  we  take  up  in 
this  article. 

1.  The  angle  between  the  axis  of  x  and  a  known  line.  Let  a 
known  line  cut  the  axis  of  x  at  the  point  L.  Then  there  are  four 
angles  formed.  To  avoid  ambiguity,  we  shall  agree  to  select  that 
one  of  the  four  which  is  above  the  axis  of  x  and  to  the  right  of 


56        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

the  line,  and  to  consider  LX  as  the  initial  line  of  this  angle.    We 
shall  denote  this  angle  by  <^.    Then  if  we  take  P  any  poiut  on 


Fig.  26 


Fig.  27 


the  terminal  line  of  ^  and  drop  the  perpendicular  MP,  we  have, 
in  the  two  cases  represented  by  figs.  26  and  27, 


tan^  = 


MP 
LM 


MP 


But  ^^^^^  is  equal  to  the  slope  of  the  line.    Therefore 
LM        ^  ^ 

tan  0  =  m. 

If  the  straight  line  is  parallel  to  OF,  <^  =  90°  and  tan  (f>  =  cc. 
If  the  line  is  parallel  to  OX,  no  angle  ^  is  formed  ;  but  since  7n  =  0, 
we  may  say  tan  <^  =  0,  whence  <^  =  0°  or  180°. 

2.  Parallel  lines.  If  two  lines  are  parallel,  they  make  equal 
angles  with  OX,  and  hence  their  slopes  are  equal.  It  follows  that 
two  equations  which  differ  only  in  the  absolute  term,  such  as 

and  Ax  +By  +  C„_  =  Q, 

represent  parallel  lines. 

More  generally,  two  straight  lines, 

and  A„x  +  B„y  +  C,  =  0, 


are  parallel  if 


^1 
J5„ 


=  0. 


ANGLES 


57 


3.  Perpendicular  lines.  Let  AB  and  CD  (fig.  28)  be  two  lines  inter- 
secting at  right  angles.  Through  P  draw  PR  parallel  to  OX  and 
let  RPD  =  <^^  and  RPB  =  (f)^.  Then  tan  <f>^  =  m^  and  tan  ^^  =  ^2. 
where  m^  and  ^/ig  ^^"^  the  slopes 
of  the  lines.    But  by  hypothesis, 

(^,=  </,,+  90°, 

whence 

1 
tan  ^2  =  —  cot  (f)^  = 

which  is  the  same  as 


tan  </)j 


Fig.  28 


That  is :    Two  straight  lines 
are    perpendicular     when     the 

slope  of  one  is  minus  the  reciprocal  of  the  slope  of  the  other.  This 
theorem  may  be  otherwise  expressed  by  saying  that  two  lines  are 
perpendicular  when  the  product  of  their  slopes  is  minus  unity. 

It  follows  that  two  straight  lines  whose  equations  are  of  the  type 


and 


Ax+Pi/+C^  =  0 
Bx  —  Ai/+C^  =  0 


are  perpendicular. 

4.  Angle  hetvjeen  two  lines.     Let  AB  and  CD  (fig.  29)  inter- 
sect at  the  point  P,  making  the  angle  BPD,  which  we  shall 

call  /3.    Draw  the  line  PR 
.D 


parallel    to    OX  and   place 
RPB  =  (f>^  and  RPD  =  (f}^. 

Then 


Fig.  29 


and  hence 

tan  yS  =  tan  {<^„  —  </>j) 

_   tan  <^„  —  tan  ^^ 
1  -|-  tan  0.,  tan  <^^ 


58        THE  POLYN^OMIAL  OF  THE  FIRST  DEGREE 

But  tan  <f>^  =  7n^  and  tan  (f)^  =  m^,  where  m^  is  the  slope  of  CD 
and  m^  is  the  slope  of  AB.    Therefore 

„       m„—m, 
tan  p  = 


1  +  m^mj 

If  <^2  is  always  taken  greater  than  ^^,  tan  yS  will  be  positive  or 
negative  according  as  /3  is  acute  or  obtuse. 

29.  Problems  on  straight  lines.  We  shall  solve  in  this  article 
certain  important  problems  which  depend  on  the  equation 

y  =  mx  +  h. 

The  essential  problem  is,  in  every  case,  to  determine  m  and  h  so 
that  the  line  will  fulfill  certain  conditions.  Since  two  quantities 
are  to  be  determined,  two  conditions  are  necessary  and  sufficient ; 
hence,  in  general,  one  and  only  one  straight  line  can  be  found  to 
satisfy  two  given  conditions. 

1.  To  find  the  equation  of  a  straight  line  which  has  a  known 
slope  and  passes  through  a  known  point.  Let  m^  be  the  known 
slope  and  P^{Xy,  y^  be  the  known  point.  Tlie  equation  of  the  line 
will  be  of  the  form  y  =  m^x  +  h,  where  h,  however,  is  unknown. 
But  the  line  contains  the  point  Py    Therefore 

y^  =  7n^x^+h, 
whence  h  =  y^—  m^Xy 

The  required  equation  is,  therefore, 

y  =  m^x  +  y^-my)c^\ 
or,  more  symmetrically, 

y-y^  =  m^{x-x^). 

Ex.  Find  the  equation  of  a  straight  line  with  the  slope  -  §  passing  tlirough 
the  point  (5,  7). 

Fir^  method.    We  have  y  =  —  §  x  +  6  ; 

then  7  =  -§(5)  +  &, 

whence  6  =  s^L. 

Therefore  the  required  equation  is 
or,  finally,  2x  +  32/-31  =  0. 


PROBLEMS  OX  STRAIGHT  LINES  59 

Second  method.    By  substituting  in  tlie  formula  we  have 

y_7  =  -|-(a;-5), 
whence  2x  +  3y  —  31  =  0,  as  before. 

2.  To  find  the  equation  of  a  straight  line  passing  through  a 
known  jpoint  and  parallel  to  a  known  line. 

The  slope  of  the  required  line  is  the  same  as  that  of  the  given 
line,  which  can  be  found  by  §  27.  Hence  the  problem  is  the  same 
as  the  preceding. 

Ex.  Find  the  equation  of  a  straight  line  passing  through  ( —  2,  3)  and  parallel 
to3x-5y  +  6  =  0. 

First  method.   The  slope  of  the  given  line  is  ^.   Therefore  the  required  line  is 
2/-3=f(x  +  2),     or    3x-5y  +  21  =  0. 

Second  method.  As  explained  in  §  28,  2,  we  know  that  the  required  equation 
is  of  the  form 

Sx-Sy  +  O^O, 

where  C  is  unknown.    Since  the  line  passes  through  (—2,  3), 

3(- 2) -5(3) +  0  =  0, 

whence  C  =  21.    Therefore  the  required  equation  is 

3x-5y  +  21  =  0, 

3.  To  find  the  equation  of  a  straight  line  passing  through  a 
known  point  and  perpendicular  to  a  known  line. 

The  slope  of  the  required  line  may  be  found  from  the  slope  of 
the  given  line,  as  in  §  28,  3.  The  problem  is  then  the  same  as 
problem  1. 

Ex.  1.  Find  a  straight  line  through  (5,  3)  perpendicular  to7x  +  9y  +  l  =  0. 

First  method.  The  slope  of  the  given  line  is  —  ^.  Therefore  the  slope  of  the 
required  line  is  |.    By  problem  1,  the  required  line  is 

2/-3  =  f(x-6),     or    9x- 72/ -24  =  0. 

Second  method.  As  shown  in  §  28,  3,  we  know  that  the  equation  of  the 
required  line  is  of  the  form  9x-7y+C  =  0.  Substituting  (5,  3),  we  find 
C  =  —  24.    Hence  the  required  line  is  9x  —  7j/  —  24  =  0. 


60        THE  POLYNOMIAL  OF  THE  FIEST  DEGKEE 

Ex.  2.  Find  the  equation  of  the  perpendicular  bisector  of  the  line  joining  (0,  5) 
and  (5,  —  11).  The  point  midway  between  the  given  points  Ls  (^,  —  3),  by  §  18. 
The  slope  of  the  line  joining  the  given  points  is  —  ^-,  by  §  27.  Hence  the  required 
line  passes  through  (§,  —  3),  with  the  slope  ^^.    Its  equation  is 

y  +  3  =  /5(x-|),     or     lOx  -  32y  -  121  =  0. 

4.  To  find  the  equation  of  a  straight  line  through  two  known 
points. 

This  problem  has  already  been  solved  in  §  26,  and  the  result 
given  in  the  form 


which  is  the  same  as 


X     y      I 

^1    2^1     1 

a^2    ^2     1 

X 

-•»!  y  -Vx 

X, 

—  X 

2       ^1-^21 

=  0, 


=  0. 


(Ex.  2,  §  3) 


Or,  by  §  27,  the  slope  of  the  required  line  is 

x^     x^ 
Hence,  by  problem  1,  the  equation  of  the  required  line  is 


Ex.    Find  a  straight  line  through  (1,  2)  and  (-  3,  6), 
By  the  formula, 

5  —  2 
y  -  ^  =  _  3  _  ^  (X  -  1),     or    3x  +  4y-ll  =  0. 

5.  To  find  the  condition  that  three  known  points  should  lie  on 
the  same  straight  line.  If  the  three  points  are  (x^,  y^),  (x^,  y^), 
and  («3,  y,),  the  condition  that  they  should  lie  on  the  same  straight 
line  is  -  - 


as  is  evident  from  4. 


x^ 

Vl 

1 

^2 

y^ 

1 

«8 

yz 

1 

=  0, 


INTEKSECTION  OF  STRAIGHT  LINES 


61 


30.  Intersection  of  straight  lines. 

Let  A^x  +  B^y  +  C;  =  0 

and  A^x  +  B^y  +  Cg  =  0 


(1) 


be  the  equations  of  two  straight  lines.  It  is  required  to  find  their 
point  of  intersection.  Since  the  coordinates  of  any  point  on  one 
of  the  lines  satisfy  the  equation  of  that  line,  the  coordinates  of  a 
point  on  both  lines  must  satisfy  both  equations  simultaneously. 
Hence  the  coordinates  of  the  point  of  intersection  of  the  lines  is 
found  by  solving  the  two  equations. 
There  are  three  cases. 


1. 


A    A 


^0. 


A       ^2 

The  solutions  are  then 


X  = 


c, 
c. 

^1 

A 
A 

^1 

^2 

y  = 


A   c, 

A       ^2 

A  A 

A      ^2 

The  two  straight  lines  intersect  in  the  corresponding  point. 


^1 
B. 


=  0,  but  at  least  one  of  the  determinants. 


B„ 


and 


not  equal  to  zero. 

The  equations  are  then  contradictory  and  the  straight  lines  do 
not  intersect.  In  fact,  §  28  shows  that  the  straight  Imes  are 
parallel. 

This  case  may  be  brought  into  connection  with  case  1  as 
follows :  In  case  1  suppose  that  A^B^  —  A„B^  is  very  small,  but 
not  zero.  The  values  of  x  and  y  are  then  very  large,  assuming 
that  the  numerators  are  not  small,  and  the  point  of  intersection 
is  then  very  remote. 


62        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

Let  now  the  lines  be  changed  in  such  a  manner  that  ^1^2—^2-^1 
approaches  zero.  The  values  of  x  and  y  increase  indefinitely,  the 
point  of  intersection  recedes  indefinitely,  and  the  lines  approach 
parallelism. 


=  0, 


C„    B„ 


=  0, 


A..     C, 


=  0. 


The  equations  are  then  not  independent  but  represent  the  same 
straight  line. 

In  this  case  the  attempt  to  use  the  solutions  as  given  in  1 
leads  to  the  indeterminate  form  ^  (§  11). 

31.  If  the  three  straight  lines 


A^oi+B^y  +  C^=Q, 
A„x  +  B^y  +  0.-,=  0, 


(1) 
(2) 
(3) 


pass  through  the  same  point,  the  three  equations  have  a  common 
solution,  and  therefore 


A 

A 

Cx 

A 

B, 

a 

A 

A 

cl 

=  0. 


(4) 


Also,  if  the  three  straight  lines  are  parallel,  the  determinant  (4) 
is  zero.     For  if  (1),  (2),  and   (3)  are  parallel,   A^B^  —  A^B^=^0, 
0,  A^Bj^—A^Bg  =  0,  and  therefore 


A^B,-A,B. 


Conversely,  if 


A 

A 

Ci 

A 

A 

c. 

A 

A 

c. 

A 

A 

Cr 

A 

A 

c. 

A 

A 

C3 

=  0. 


=  0, 


the  Imes  (1),  (2),  and  (3)  either  pass  through  the  same  point  or  are 
parallel.  For,  by  §  7,  if  two  oi'  the  lines  intersect,  the  coordinates 
of  the  point  of  intersection  satisfy  the  other. 


DISTANCE  FROM  A  STRAIGHT  LINE 


63 


32.  Distance  of  a  point  from  a  straight  line. 

Take  the  equation  of  any  straight  line,  written  in  the  form 

y  —  mx  —  b  =  0,  (1) 

and  consider  the  polynomial 

y  —  mx  —  h,  (2) 

which  stands  upon  the  left-hand  side.    We  may  substitute  in  (2) 
the  coordinates  (x^,  y^  of  any  point  P^,  and  thus  obtain  a  value 
of  (2)  which  is  zero  when  P^  lies 
on  the  line  (1),  but  not  other- 
wise.   We  wish  now  to   obtain 
the  meaning  of 

y^—  mx^ —  h 

when  P^  is  not  on  (1).    For  that 

purpose,  let  LK  (fig.  30)  be  the 

line  (1),  and  let  MP^,  the  ordinate  j^,g  30 

of  ij,  cut  LK  in  Q.    Then  the 

abscissa  of  Q  is  x^  and  its  ordinate  is  MQ.    From  (1) 

MQ  =  mx^-\-  h. 
Hence  y^  —  ma;^  —  ^  =  2/i  ~  (^^i  +  ^) 

=  MPi-MQ  =  QP^. 

It  is  clear  that  y^  —  inx^  —  &  is  a  positive  quantity  when  {x^,  y^ 
lies  above  the  hne  LK,  and  is  a  negative  quantity  when  {x^,  y^  lies 
below  LK.  It  is  also  evident  from  the  triangle  P^QR,  and  from 
a  like  triangle  in  other  cases,  that  the  length  of  P^R  is  numeric- 
ally equal  to  P^Q  cos  <^.    But  tan  <^  =  m,  and  hence 


We  have,  then. 


cos<^ 
P^R 


mx^ —  h 


±ViT 


m 


64        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 


We  may,  if  we  wish,  always  choose  the  +  sign  in  the  denomi- 
nator. Then  P^R  is  positive  when  ij  is  above  y  =  mx  +  h,  and 
negative  when  i^  is  below. 

If  the  equation  of  the  straight  line  is  in  the  form 

Ax  +  By+C=Q, 

A  C 

m= and  &  = Therefore 

B  B 


and 


Ax^  +  By^  +C=B{y^—  mx^  —  5), 
Ax,  +  By,+  C 

y/A'  +  B^ 


It  appears,  then,  that  the  polynomial  Ax^  +  By^  +  C  and  the  per- 
pendicular PyR  are  positive  for  all  points  on  one  side  of  the  line 
Ax-\-By  -\-C=  0,  and  negative  for  all  points  on  the  other  side. 
To  determine  which  side  of  the  line  corresponds  to  the  positive 
sign,  it  is  most  convenient  to  test  some  one  point,  preferably  the 
origin. 

33.  Normal  equation  of  a  straight  line.  Let  LK  (fig.  31)  be  any 
straight  line  and  let  OD  be  the  normal  (or  perpendicular)  drawn 
from  the  origin.  Let  the  length  of  OD  be  p  and  let  the  angle 
XOD  be  a.    Take  P  any  point  on  LK.    The  projection  of  OP  on 

OD  is  equal  to  the  sum  of  the 
projections  of  OM  and  MP  (§  15). 
But  the  projection  of  OP  on  OD 
is  p,  since  ODP  is  a  right  angle. 
The  projection  of  OM  on  OD  is 
xcoBa  (§  14),  and  that  of  3IP  is 
y  cos  (a  —  90°)  =  2/  sin  a.     Hence 


Fig.  31 


or 


p  =  x  cos  a  +  y  sin  a, 
X  cos  a-\-  y  sin  a  —  j?  =  0. 


This  equation,  being  true  for  the  coordinates  of  any  point  on 
LK  and  for  those  of  no  other  point,  is  the  equation  of  LK,  It  is 
called  the  normal  equation  of  a  .straight  line. 


NORMAL  EQUATION  65 

Since  sin'^a;  +  cos^a  =  1,  it  follows  from  §  32  that 
Xj^  cos  a  +  2/j  sin  a  —  p 
is  numerically  equal  to  the  distance  of  {x^,  y^  from 
X  cos  a  +  2/  sin  a;  —  ^  =  0. 
It  is  sometimes  desirable  to  change  an  equation 

Ax  +By  +  C  =  0 
into  X  cos a  +  2/sina  —  jt?  =  0. 

For  that  purpose  it  is  enough  to  notice  that  since  any  value  of 
{x,  y)  which  satisfies  one  equation  must  satisfy  the  other,  the  one 
is  a  multiple  of  the  other.    Hence 

J  A  =  k  cos  a,     B  =  k  sin  a,    C  =  —  ly, 

where  k  is  an  unknown  factor.    But  from  these  last  equations  we 
have 

A'  +  B''  =  k^. 

Therefore  cos  a  = 


sma  = 


p  = 


±^A'  +  B'' 
B 


-c 


±y/A'  +  B- 


Since  p  is  to  be  positive,  the  sign  of  the  radical  must  be  oppo- 
site to  that  of  C. 

PROBLEMS 
Plot  the  graphs  of  the  following  equations  : 

1.  5a; -3?/ +  10  =  0.  3.  a;  +  3?/ -  7  =  0.  5.  3x  +  52/  =  0. 

2.  4x  +  6?/  +  12  =  0.  4.  2x-9y  =  0.  6.  4a;  +  7  =  0. 

7.  .5y-8  =  0. 

8.  Two  numbers  are  to  be  found  such  that  one  half  of  one  plus  one  third 
of  the  other  is  eciual  to  unity.  Show  how  one  number  may  be  graphically 
found  when  the  other  is  known. 


66        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

9.  A  plane  figure  is  in  the  form  of  a  square,  3  ft.  on  one  side,  surmounted 
by  a  triangle  constructed  on  one  of  its  sides  as  a  base.  Express  the  area  of  the 
above  figure  in  tenns  of  the  altitude  of  the  triangle,  and  plot  the  graph  of  the 
function. 

10.  Express  the  number  of  inches  in  any  length  as  a  function  of  the  number 
of  centimeters,  and  express  the  same  as  a  graph. 

11.  A  uniform  elastic  string  of  length  I  is  subjected  to  a  stretching  force/. 
If  V  is  the  new  length,  l'=l(l  +  mf),  where  m  is  a  constant.  Plot  the  graph, 
showing  the  relation  between  I'  and  /. 

12.  K  t  represents  the  boiling  point  in  degrees  Centigrade  at  a  height  h  in 
meters  above  sea  level,  then  approximately  h  =  295  (100  —  t).   Plot  the  graph. 

13.  The  pressure  on  a  square  unit  of  horizontal  surface  immersed  in  a  liquid 
is  equal  to  the  weight  of  the  column  of  liquid  above  it.  Express  the  pressure  at 
a  depth  x  below  the  surface  of  a  body  of  water,  the  density  of  the  water  being 
taken  as  unity.  Express  also  the  pressure  x  units  below  the  surface  of  a  body 
of  water  over  which  is  a  body  of  oil  of  density  .9  and  of  depth  8  units.  Plot 
the  graphs. 

14.  A  road  starts  at  an  elevation  of  100  ft.  above  sea  level  and  has  a  uniform 
up  grade  of  15  per  cent ;  i.e.  it  rises  15  ft.  in  every  100  ft.  of  horizontal  length. 
Express  the  distance  above  sea  level  on  the  road  as  a  function  of  the  horizontal 
distance  from  the  point  of  departure,  and  construct  the  graph. 

15.  A  tank  of  water  contains  100  gal.  A  tap  is  opened,  causing  the  water 
to  flow  out  at  a  uniform  rate  of  §  gal.  per  minute.  Express  the  amount  of 
water  in  the  tank  as  a  function  of  the  time,  and  construct  the  graph. 

16.  Find  the  equation  of  the  straight  line  of  which  the  slope  is  7  and  the 
intercept  on  OY  is  —  3. 

17.  Find  the  equation  of  the  straight  line  passing  through  the  point  (0,  —  3) 
and  making  an  angle  of  135°  with  OX. 

18.  y  ind  the  equation  of  a  straight  line  making  an  angle  of  60°  with  OX  and 
cutting  off  an  intercept  —  5  on  OY. 

19.  A  straight  line  making  a  zero  intercept  on  OY  makes  an  angle  of  120° 
with  OX.   Find  its  equation. 

20.  A  straight  line  making  a  zero  angle  with  OX  cuts  OY  at  a  point  5  units 
from  the  origin.    Find  its  equation. 

21.  Find  the  acute  angle  between  the  lines  2x-3y+5  =  0  an^  x+2y  +  2  =  0. 

22.  Find  the  acute  angle  between  the  lines  2x  +  Sy  -  0  =  0  and  2x  +  y+l  =  0. 

23.  Find  the  acute  angle  between  the  lines  4x  +  y  —  2  =  0  and  3x  +  oy  +  S  =  0. 

24.  Show  that  2x  +  14i/  —  17  =  0  bisects  one  of  the  angles  between  the  lines 
8z4-6y-ll  =  0  andSx  -4y +  3  =  0. 

25.  Find  the  equation  of  the  straight  line  through  the  point  (-  4,  5)  parallel 
to  the  line  6x  —  iy+l  =  0. 


PROBLEMS  67 

26.  Find  the  equation  of  the  straight  line  through  (3,  —  1)  parallel  to  the  line 

X  -  2/  =  8. 

27.  Find  the  equation  of  the  straight  line  through  the  point  (2,  —  11)  per- 
pendicular to  the  line  9x  —  8y  +  6  =  0. 

28.  Find  the  equation  of  the  straight  line  through  the  origin  perpendicular 
to  the  line  6x  +  5?/—  3  =  0. 

29.  Find  the  equation  of  the  straight  line  through  the  points  (—  2,  —  3) 
and  (0,  4). 

30.  Find  the  equation  of  the  straight  line  through  the  points  (2,  —  1) 
and  (3,  2). 

31.  Find  the  equation  of  the  straight  line  through  the  points  (—1,  3) 
and  (—1,  5). 

32.  Find  the  angle  between  the  straight  lines  drawn  from  the  origin  to  the 
points  of  trisection  of  that  part  of  the  line  6x  +  iy  =  24:  which  is  included 
between  the  coordinate  axes, 

33.  Find  the  equation  of  the  perpendicular  bisector  of  the  line  joining 
(-3,  6)  and  (-4,  1). 

34.  A  straight  line  is  perpendicular  to  the  line  joining  the  points  (—4,  —  2) 
and  (2,  —  6)  at  a  point  one  third  of  the  distance  from  the  first  to  the  second 
point.    AVhat  is  its  equation  ? 

35.  Find  the  equation  of  the  straight  line  through  (3,  5)  parallel  to  the 
straight  line  joining  (2,  5)  and  (—  6,  —2). 

36.  Find  the  equation  of  the  straight  line  parallel  to  the  line  2z  —  3^  +  5  =  0 
and  bisecting  the  straight  line  joining  (—  1,  2)  and  (4,  5). 

37.  Find  the  equation  of  the  straight  line  perpendicular  to  3x  —  5^  =  9  and 
bisecting  that  portion  of  it  which  is  included  between  the  coordinate  axes. 

38.  What  is  the  equation  of  a  straight  line  the  intercepts  of  which  on  the 
axes  of  X  and  y  are  2  and  —  5  respectively  ? 

39.  What  is  the  equation  of  the  straight  line  the  intercepts  of  which  on  the 
axes  of  X  and  y  are  —  4  and  —  7  respectively  ? 

40.  In  the  triangle  A  {-  2,  -  2),  B  {1,  -  8),  C  (0,  -  7),  a  straight  line  is 
drawn  bisecting  the  adjacent  sides  AB&nd  BC.  Prove  that  it  is  parallel  to  ^C 
and  half  as  long. 

41.  Find  the  equation  of  a  straight  line  through  (4,  ^)  and  the  point  of 
intersection  of  the  lines  3x  —  4y  —  2  =  0  and  12x  —  loy  —  8  =  0. 

42.  Find  the  equation  of  the  straight  line  passing  through  the  point  of  inter- 
section of  X  —  22/  —  5=0  and  2x  —  3y  —  8  —  0  and  parallel  to3x  —  2?/  +  2  =  0. 

43.  Find  the  equation  of  the  straight  line  through  the  point  of  intersection 
of6x  —  2y  —  11  =  0  and  4x  —  Cy—  5  =  0  and  perpendicular  to4x— 2/  +  l  =  0. 


68        THE  POLYNOMIAL  OF  THE  FIRST  DEGREE 

44.  Find  the  equation  of  the  straight  line  joining  the  point  of  intersection 
of  the  lines  2x  —  y  +  o  =  0  and  x+  y  +  1  —  0  and  the  point  of  intersection  of 
the  lines  x  —  y  —  7  =  0  and  2x  +  y  —  o  =  0. 

45.  Determine  the  value  of  m  so  that  the  line  y  =  mx  +  3  shall  pass  through 
the  point  of  intersection  of  the  lines  y  =  2x  +  1  and  y  =  x  +  o. 

46.  Find  the  vertices  and  the  angles  of  the  triangle  formed  by  the  lines 
x  =  0,x-y  +  2  =  0,  and  2x  +  32/-21  =  0. 

47.  Find  the  distance  of  (3,  5)  from  the  line  y  =  4  x  -  8.  On  which  side  of 
the  line  is  the  point  ? 

48.  How  far  distant  from  the  line  2x  +  Sy  +  8  =  qis  the  point  (7,  -  4),  and 
on  which  side  of  the  line  is  it  ? 

49.  Find  the  distance  from  the  point  (6,  —  a)  to  the  line  -  +  -  =  1 

a      b 

50.  The  base  of  a  triangle  is  the  straight  line  joining  the  points  (-  1,  3)  and 
(5,  -  1).    How  far  is  the  third  vertex  (6,  -  2)  from  the  base  ? 

51.  The  vertex  of  a  triangle  is  the  point  (6,  -  2)  and  the  base  is  the  straight 
line  joining  (-  3,  2)  and  (4,  3).    Find  the  lengths  of  the  base  and  the  altitude. 

52.  Find  the  distance  between  the  two  parallel  lines  4x  +  Sy  -10  =  0  and 
4x  +  3  2/  -  8  =  0. 

■  53.  A  straight  line  is  7  units  distant  from  the  origin  and  its  normal  makes 
an  angle  of  30°  with  OX.    What  is  its  equation  ? 

54.  The  normal  to  a  straight  line  which  is  5  units  distant  from  the  origin 
makes  the  acute  angle  tan-i  J  with  OX.    What  is  the  equation  of  the  line  ? 

55.  A  straight  line  4  units  distant  from  the  origin  makes  an  angle  of  45° 
with  OX.    What  is  its  equation  ? 

56.  The  normal  to  a  straight  line  makes  an  angle  tan-i|with  OX.  The 
line  passes  through  the  origin.    What  is  its  equation  ? 

57.  The  normal  to  a  straight  line  makes  an  angle  of  90°  with  OX.  The 
line  is  7  units  distant  from  the  origin.    What  is  its  equation  ? 

58.  Find  a  point  on  the  line  4  x  +  3  y  =  12  equidistant  from  the  point" 
(-1,  -2)  and  (1,4). 

59.  Find  the  equation  of  the  perpendicular  bisector  of  the  base  of  an 
isosceles  triangle  having  its  vertices  at  the  points  (3  2)  (-2  -3^  and 
(2,  -  5).  V  '     /'   V        .         /, 

60.  A  point  is  equally  distant  from  (2,  1)  and  (-  4,  3),  and  the  slope  of  the 
straight  line  joining  it  to  the  origin  is  §.    Where  is  the  point  ? 

61.  A  point  is  7  unite  distant  from  the  origin,  and  the  slope  of  the  straight 
hue  joining  it  to  the  origin  is  §.    What  are  ite  coordinates  ? 

62.  Perpendiculars  are  let  fall  from  the  point  (.5,  0)  upon  the  sides  of  the 
triangle  tlie  vertices  of  which  are  at  the  points  (4,  3),  (-  4,  3)    and  (0    -  5) 
Show  that  the  feet  of  the  three  perpendiculai-s  lie  on  a  straight  line 


PEOBLEMS  69 

63.  Find  a  point  on  the  line  x  +  2  y  —  3  =  0,  the  distance  of  which  from  the 
axis  of  X  equals  its  distance  from  tlie  axis  of  y. 

64.  One  diagonal  of  a  parallelogram  joins  the  points  (4,  —  2)  and  (—  4,  —  4). 
One  end  of  the  other  diagonal  is  (1,  2).    Find  its  equation  and  length. 

65.  Find  the  equations  of  the  straight  lines  through  the  point  (—2,  0) 
making  an  angle  tan-i  |  with  the  line  3x  +  4y  +  6=:0. 

66.  Find  the  equations  of  the  straight  lines  through  (2,  2)  making  an  angle 
of  45°  with  the  line  3  x  -  2  ?/  =  0. 

67.  Find  the  equations  of  the  straight  lines  through  the  point  (2,  1)  making 
an  angle  tan-i  ^  with  the  line  2x  —  y  —  3=0. 

68.  Derive  the  equation  of  the  straight  line  making  the  intercepts  a  and  h 
on  the  axes  of  x  and  y  respectively. 

69.  Prove  analytically  that  the  locus  of  points  equally  distant  from  two 
points  is  the  perpendicular  bisector  of  the  straight  line  joining  them. 

70.  Prove  analytically  that  the  medians  of  a  triangle  meet  in  a  point. 

71.  Prove  analytically  that  the  perpendicular  bisectors  of  the  sides  of  a  tri- 
angle meet  in  a  point. 

72.  Prove  analytically  that  the  perpendiculars  from  the  vertices  of  a  tri- 
angle to  the  opposite  sides  meet  in  a  point. 

73.  Prove  analytically  that  the  perpendiculars  from  any  two  vertices  of  a 
triangle  to  the  median  from  the  third  vertex  are  equal. 

74.  Prove  analytically  that  the  straight  lines  joining  the  middle  points  of 
the  adjacent  sides  of  any  quadrilateral  form  a  parallelogram. 

75.  Prove  analytically  that  the  straight  lines  drawn  from  a  vertex  of  a  paral- 
lelogram to  the  middle  points  of  the  opposite  sides  trisect  a  diagonal. 


CHAPTER  IV 
THE  POLYNOMIAL  OF  THE  JNTth  DEGREE 


34.  Graph  of  the  polynomial  of  the  second  degree. 

The  polynomial  of  the  second  degree  is  aa^+hx  +  c.  Its 
graph  may  be  plotted  by  equating  it  to  y  and  proceeding  as  in 
§§  20  and  23. 

Ex.  1.    a;2  +  2x  +  2. 

Place  y  =  x^+2x  +  2  and  assume  integral 
values  of  x.  The  corresponding  values  of  x  and 
y  are  given  in  tlie  following  table  : 

y 


X 

y 

- 1 

1 

-2 

2 

-3 

5 

-4 

10 

-5 

17 

As  in  §  20,  we  plot  these  points  (fig.  32),  and 
are  then  to  draw  a  smooth  curve  through  them. 

But  we  notice  that  these  points  are  nearer 
together  in  some  places  than  in  others.  It  follows 
that  in  some  parts  the  curve  would  be  more 
accurate  than  in  others.  To  obviate  this  diffi- 
culty we  assume  such  fractional  values  of  x 
as  will  locate  points  between  the  more  widely 
separated  points  already  plotted. 

We  thus  form  the  table  : 


x 

y 

1.5 

7.3 

2.5 

13.3 

3.5 

21.3 

-2.5 

3.3 

X 

y 

-3.5 

7.3 

-4.5 

13.3 

-5.5 

21.3 

Fig.  32 


Plotting  these  points  also,  and  drawing  the  curve,  we  have  (fig.  32)  the  graph  of 
the  given  polynomial,  a;^  +  2x  +  2.  The  graph  lies  entirely  above  the  axis  of  x, 
and  recedes  constantly  from  it  as  x  increases  numerically,  since  the  polynomial 
is  positive  for  all  values  of  x,  and  increases  in  value  as  x  increases. 

70 


POLYNOMIAL  OF  THE  SECOND  DEGREE 


71 


Ex.  2.    2x2  +  x  -6. 

Place  y  =  2x^  +  X  —  Q  and  assume  integral  values  of  x 

Hence  the  table : 


y 

-6 

-3 

4 

15 


X 

-  1 
-2 
-3 


y 


On  plotting  these  points  (fig.  33)  we  see  that  it 
is  desirable  to  assume  fractional  values  of  x. 
Hence  the  table : 


X 

y 

1.5 

0 

2.3 

6.9 

2.6 

10.1 

-1.5 

-3 

-2.5 

4 

X 

y 

-3.3 

12.5 

-3.7 

17.7 

-    .5 

-6 

-    .3 

-6.1 

-    .7 

-5.7 

Fio.  33 


In  obtaining  this  new  set  of  points  we  have 
assumed  —  .5,  —  .3,  —  .7  as  values  for  x,  with  the 
aim  of  locating  as  closely  as  possible  the  turning 
point,  or  vertex,  as  it  will  be  called,  of  the  curve. 
Plotting  these  points  also,  we  draw  the  curve  (fig.  33). 

It  is  especially  to  be  noted  that  the  curve  cuts  the 
axis  of  X  when  x  =  —  2  and  when  x  =  1.5.    But  these  two  values  of  x,  since  they 
make  2  x'^  +  x  —  0  equal  to  zero,  are  the  roots  of  the  equation  2  x^  +  x  —  6  =  0. 

As  the  graph  of  the  polynomial  in  Ex.  1  did  not  intersect  the  axis  of  x,  we 
conclude  that  the  equation  formed  by  placing  it  equal  to  zero  has  no  real  roots. 
Solving  that  equation  we  find  that,  in  fact,  the  roots  are  —  1  ±  V—  1. 

35.  Let  us  now  consider  the  general  polynomial  of  the  second 
degree,  ax^+hx  +  c,  of  which  the  two  polynomials  just  plotted 
are  special  cases. 

If  we  place  y  =  ax^  +  hx  +  c,  we  can  write 


7/  =  a 


0      b  cl 

3y'+-x+-\ 
a  aj 

h  y    c 

^-  +  —    +  - 
2  a/       a 


¥—  4ac 
4  a' 


+    x-h 


h 


72    THE  POLYNOMIAL  OF  THE  Nth   DEGREE 

-4 
4:  a' 


The  expression  in  brackets  is  the  constant, -— ^ —  >  plus 


a  function  of  x,  {x  +  -^-]>  which  is  always  positive  except  for 

h  V        2  a/ 

x  =  —  ——,  when  it  is  zero. 
2  a 

At  first  we  shall  regard  a  as  positive.    It  follows  that  y  has  its 

least  value  when  x  = Therefore  the  lowest  point,  the  vertex, 

2  a 

of  the  curve  will  be  ( > )  •    As  values  greater  and 

\     2  a  4a      / 

less  than are  assigned  to  x,  x  +  - —  increases  numerically, 

2  a  2  a 

y  increases,  and  the  corresponding  point  of  the  curve  rises  in 

the  plane.    Moreover,  if  x  is  assigned  the  values  — 1-  k  and 

h 

A;,  A;  being  any  assumed  constant  value,  the  corresponding 

2  a 

values  of  y  are  the  same.    Hence  the  curve  is  symmetrical  with 

respect  to  the  straight  line  x  =  —  - —  >  which  line  passes  through 

the  vertex  of  the  curve  parallel  to  the  axis  of  y. 

If  a  is  negative,  it  can  be  proved  in  the  same  way  that  the  curve 

has  an  axis  of  symmetry,  a;  =  —  - —  >  which  passes  through  its 

vertex,  which  is  in  this  case  the  highest  point  of  the  curve. 

36.  Now  that  we  have  proved  that  the  graphs  of  all  quadratic 
polynomials  in  x  are  alike,  having  a  vertex  and  an  axis  of  sym- 
metry passing  through  it,  we  can  plot  them  more  easily  than  was 
possible  before,  as  is  shown  by  the  two  following  examples. 

Ex.  1.    4z2_4x  +  l. 
I  y  =  4a;2_4x  +  l  =  4(x2_x  + i)  =  4(a;-^)2. 

I  Therefore  the  vertex  of  the  graph  is  (^,  0),  and  the 

/  axis  of  symmetry  is   the   line  x=  ^.    Beginning  with 

/  the  value  ^,  we  assign  to  x  values  greater  and  less 

/  than  ^,   thereby  locating  points  on  both  sides  of  the 

axis  of  symmetry,  and  plot  the  graph  which  is  repre- 
sented in  fig.  34. 

V 

We  see  that  the  equation  4x2_4x  +  l  =  0  has  two 

equal  real  roots,  the  graph  being  tangent  (§  37)  to  the 
Fig.  34  axis  of  x  at  the  point  x  =  \. 


DISCRIMINANT 


73 


Ex.2.     -2x2  +  3x. 

y  =  -2x2  +  3x 
=  -2(x2-|x) 
=  -2[{x- 1)2-^9^]. 

Therefore  the  vertex  of  the  graph  is  (|,  |)  and 
the  axis  of  symmetry  is  the  line  x  =  |.  The  graph 
is  represented  in  fig.  35.  We  see  that  it  crosses 
the  axis  of  z  at  two  different  points.  Hence  the 
equation  — 2x2  +  3x=0  has  two  unequal  real 
roots,  which  are  found  to  be  0  and  f . 


Fid.  35 


37.  Discriminant  of  the  quadratic  equation.    Turning  now  to 

the  constant —  in  the  equation 

4a^ 


y  =  a 


h^—4ac 
4  a' 


+    x  + 


2a 


of  §  35,  we  have  three  cases  to  consider. 

1.  If  &'—  4  ac  >  0,  the  vertex  of  the  graph  is  below  the  axis  of 
X  when  a  >  0,  and  above  the  axis  of  x  when  a  <  0,  and  accord- 
ingly the  graph  intersects  the  axis  of  x  in  two  points. 

2.  If  &'—  4  ac  =  0,  the  vertex  of  the  graph  is  on  the  axis  of  x, 
and  hence  the  graph  intersects  the  axis  of  a;  in  a  single  point. 

3.  If  b'^—4:ac  <  0,  the  vertex  of  the  graph  is  above  the  axis 
of  X  when  a  >  0,  and  below  the  axis  of  x  when  a  <  0,  and  the 
graph  does  not  intersect  the  axis  of  x  at  all. 

Now  let  us  suppose  that  different  values  are  assigned  to  the 
constants  a,  h,  and  c,  in  such  a  way  as  to  make  &^—  4  ac  decrease, 
beginning  with  a  positive  value.  Then  the  vertex  of  the  graph 
rises  or  falls  in  the  plane  until,  when  J^—  4  ac  =  0,  it  lies  on  the 
axis  of  x.  At  the  same  time,  the  points  in  which  the  graph  inter- 
sects the  axis  of  x  have  been  approacliing  each  other,  and  finally 
coincide,  when  the  graph  is  said  to  be  tangent  to  the  axis  of  x. 

Eecalling  that  the  abscissas  of  the  points  of  the  graph  on  the 
axis  of  X  are  the  real  roots  of  the  equation  formed  by  placing  the 
expression  equal  to  zero,  we  can  tabulate  the  following  results. 


74 


THE  POLYNOMIAL  OF  THE  iS^TH  DEGKEE 


1.  If  &^  —  4  ac  >  0,  the  graph  of  a^^  +hx  +  c  intersects  the  axis 
of  X  at  two  points,  and  the  equation  aa^  -\-bx-i-  c  =  0  has  two  real 
roots,  which  are  unequal. 

2.  If  6^—  4  ac  =  0,  the  graph  of  aa^+  hx-\-  c  is  tangent  to  the 
axis  of  X,  and  the  equation  aa?+hx  +  c  =  0  has  two  real  roots, 
which  are  equaL 

3.  If  &^—  4  ac  <  0,  the  graph  of  ax^-\-  hx  +  c  is  entirely  on  one 
side  of  the  axis  of  x,  and  the  equation  ai?  +  Jx  +  c  =  0  has  only 
imaginary  roots. 

The  expression  J^  —  4  ac  is  called  the  discriminant  of  the  quad- 
ratic equation,  as  its  sign  indicates  the  nature  of  the  roots  of  the 
equation. 
Y  38.  Graph  of  the  polynomial  of  the  nth  degree. 

Let  the  polynomial  be 

a^x"-\-  a^af-'+  a^3(f-'^-\ 1-  a^_^x+a^. 

In  general  this  polynomial  contains  n  +  1  terms. 
If  any  term  is  lacking,  we  may  consider  that  its 
coefficient  lias  become  zero. 

We  will  begin  by  plotting  the  graphs  of  some 
5^  special  numerical  cases. 


Ex.  1.    z3. 

Place  y  =  x^  and  assume  values  of  x.    Hence  the  table ; 


Fio.  36 


X 

y 

0 

0 

1 

1 

2 

8 

-  1 

-  1 

-  2 

-  8 

.5 

.1 

-    .5 

-    .1 

z 

y 

1.5 

3.4 

-  1.5 

-     3.4 

2.3 

12.2 

-  2.3 

-  12.2 

2.7 

19.7 

-  2.7 

-  19.7 

Drawing  a  smooth  curve  through  these  points,  we  have  the  curve  of  fig.  36. 
It  is  called  a  cubical  parabola. 


EXAMPLES  OF  GRAPHS 


76 


Ex.  2.    X*. 

Place  y  =  X*  and  assume  values  of  x.    Hence  the  table ; 


X 

y 

0 

0 

1 

1 

2 

16 

-1 

1 

-2 

16 

.5 

.1 

.7 

.2 

.8 

.4 

.9 

.7 

1.1 

1.6 

1.3 

2.9 

1.6 

6.1 

X 

y 

1.7 

8.4 

1.9 

13.0 

-    .5 

.1 

-    .7 

.2 

-    .8 

.4 

-    .9 

.7 

-1.1 

1.6 

-1.3 

2.9 

-1.5 

5.1 

-1.7 

8.4 

-1.9 

13.0 

The  curve  is  represented  in  fig.  37. 


Ex.  3.    x5. 

Place  7/  =  x^  and  assume  values  of  x. 
Hence  the  table  : 


Fig.  37 


-X 


X 

y 

0 

0 

1 

1 

2 

32 

-  1 

-    1 

-2 

-32 

.7 

.2 

.9 

.6 

1.2 

2.6 

1.4 

5.4 

1.6 

10.6 

1.7 

14.2 

X 

y 

1.8 

18.9 

1.9 

24.8 

-    .7 

-      .2 

-    .9 

-      .6 

-1.2 

-    2.5 

-1.4 

-    5.4 

-1.6 

-10.5 

-1.7 

-  14.2 

-  1.8 

-  18.9 

-  1.9 

-24.8 

Fig.  38 


The  curve  is  represented  in  fig.  38. 

In  each  of  the  three  examples  above,  the  curve  crossed 
the  axis  of  x  at  the  origin,  and  the  corresponding  equation 
had  the  root  zero. 


76 


THE  POLYNOMIAL  OF  THE  Nth   DEGREE 


Ex.  4.   x^  -2x-  +  ox  -6. 

Place    y  =  x^  —  2 x^  +  Sx  —  a    and    assume    values   of    x. 
Hence  the   table : 


x 

y 

0 

-   6 

1 

-   4 

2 

0 

3 

12 

-1 

-12 

Fig.  39 


-  2  i  -  28 

The  curve  is  represented  in  fig.  39. 

This  curve  crosses  the  axis  of  x  at 
the  point  x  =  2,  and  hence  the  equation 
x^  —  2x2  +  3x-6  =  0  has  2  for  a  real  root. 
Its  other  roots  are  imaginary,  i.e.  ±  V—  3. 

Ex.  5.   4x8  +  4x2-  9x  -  9. 

tlace  y  =  4x'  +  4x2  _  9x  —  9  and  assume 
values  of  x.    Hence  the  table : 


X 

y 

1.5 

-    2.6 

2.5 

4.6 

2.7 

7.2 

-1.5 

-18.4 

-1.7 

-21.8 

X 

y 

0 

-  9 

1 

-10 

2 

21 

-1 

0 

-2 

-  7 

X 

y 

1.6 

0 

1.3 

-5.2 

1.7 

6.9 

-  .5 

-4.0 

-1.5 

0 

-1.3 

.7 

This  curve  is  represented  in  fig.  40.  It  crosses  the  axis  of 
X  at  three  points,  —  when  x  =  1.5,  when  x  =  —  1.5,  and  when 
x  =  —  l.  Hence  ±1.5  and  —  1  are  real  roots  of  the  equation 
4x8  +  4x2 -9x -9  =  0. 

Without  discussing  any  more  numerical  examples 
we  can  see  that,  in  general,  the  abscissas  of  the  points 
on  the  axis  of  x  of  the  graph  of  the  polynomial 


Fig.  40 


are  real  roots  of  the  equation 

a^of+  a^x"-'+a^oif-'+  ...  +  a„_,x+  a„  =  0. 


SOLUTION  BY  FACTORING  77 

Conversely,  the  real  roots  of  the  equation 

a(,af  +  a^af~^+  a^o(f~^-\-  •  •  •  +  a„_ia;+  «„  =  0 
are  the  abscissas  of  the  points  at  which  the  graph  of 

ftpaf  +  a^x''~^+  a^3(f*~^+  •  •  •  +  a„_jiz;  +  a„ 

intersects  the  axis  of  x,  for  they  make  y  =  0. 

Moreover,  if  the  graph  of  the  polynomial  does  not  intersect 
the  axis  of  x,  the  corresponding  equation  has  no  real  roots;  and 
conversely,  if  the  equation  has  no  real  roots,  the  graph  of  the 
polynomial  does  not  intersect  the  axis  of  x. 

39.  Solution  of  equations  by  factoring.  Let  f{x)  be  a  poly- 
nomial which  can  be  separated  into  factors  fy{x),  fj^x),  f^{x),  •  •  • , 
each  of  which  is  necessarily  of  lower  degree  than /(a?).  Then  the 
equation 

/(^)=0  (1) 

may  be  written  in  the  form 

A{^)-U^)-U^)'--  =  ^-  (2) 

It  is  evident  that  any  value  of  x  which  makes  one  of  the  fac- 
tors f-^{x),  f^(x),  fJx),  ■  •  •  zero,  satisfies  equation  (2),  and  hence 
equation  (1),  i.e.  is  a  root  of  equation  (1).  But  such  a  value  of  x 
is  evidently  a  root  of  some  one  of  the  equations 

Conversely,  any  root  of  equation  (1)  must  satisfy  equation  (2), 
and  hence  must  make  some  one  of  the  factors  /^{x),  f^(x),  fj^x),  •  •  • 
zero  ;  for  if  no  one  of  these  factors  is  zero,  their  product  cannot  be 
zero.  Hence  the  solution  of  the  equation  f{x)  =  0  is  reduced  to 
the  solution  of  the  separate  equations 

/i(^)=0,        /.(^)=o,       /3(^)=0, 

In  applying  this  method  it  is  usually  desirable  to  have  no  fac- 
tor of  higher  degree  than  the  second ;  but  there  is  no  advantage 
in  carrying  the  factoring  any  further,  as  any  quadratic  equation 
can  be  readily  solved. 


78    THE  POLYNOMIAL  OF  THE  Ath  DEGREE 

Ex.  1.    Solve  the  equation  x^  =  8. 

By  transposition,  x'^  —  8  =  0 ; 

whence,  by  factoring,  (x  —  2)  (x^  +  2  a;  4-  4)  =  0. 

.-.  x-2  =  0  or  x2  +  2x  +  4  =  0; 
whence  x  =  2    or     —  1  ±  V—  3. 

Since  the  original  equation  might  have  been  written  x  =  '^S,  we  see  that  the 
three  values  of  x  which  have  been  found  are  each  a  cube  root  of  8.  In  fact, 
every  number  has  three  cube  roots,  which  may  be  found  by  solving  the  equation 
formed  by  placing  x^  equal  to  the  number. 

Ex.  2.    Solve  the  equation  x*  +  9  =  0. 
This  equation  may  be  written 

(x*  + 6x2 +  9) -6x2  =  0; 

whence,  by  factoring,  (x2  +  Vo  x  +  3)  (x2  —  V6  x  +  3)  =  0. 

.-.  x2  +  Vox  +  3  =  0,     or    x2  -  V6x  +  3  =  0 ; 

-  V6  ±  V3^  Ve  +  V^Te 

whence  X  —  — = or    —= - . 

2  2 

It  is  to  be  noted  that  every  number  has  four  fourth  roots,  which  may  be  found 
by  a  method  similar  to  that  suggested  above  for  linding  its  three  cube  roots. 

40.  Factors  and  roots.  It  follows  immediately  from  the  pre^ 
ceding  article  that  if  x  —  r  is  a  factor  of  f{x),  then  r  is  a  root  of 
the  equation  f{x)  =  0. 

Conversely,  if  r  is  a  root  of  the  equation  f(x)=0,  then  the 
polynomial  f  {x)  is  divisible  hy  x  —  r. 

Let         f{x)  =  a^x^^  a^af -'  +  •  •  •  +  a,^_^x  +  a„, 
and  let  r  be  a  root  of  f{x)  =  0.    Then 

f{r)  =  a.,r^  +  a^r'-i  +  .  .  .  +  a^_^r  +  a„  =  0. 
r,f{x)=f{x)~f(r) 

_       =  (a^aj"  +  a^a;"-'  +  •  •  •  +  rr.,,,,  a?  +  aj 

-  («o^"  +  «i^"~'  H +  «„_ir  +  a„) 

=  a„(af -  r")  +  a^(a--'  -  r'-^)  +  •  •  •  +  a„_>(.t^  -  r). 

As  f{x)  is  expressed  as  a  series  of  terms  each  of  which,  being 
the  difference  of  the  same  positive  integral  powers  of  x  and  r,  is 
divisible  by  x-r,  it  follows  that /(a-)  is  divisible  by  a;-r. 


FACT0E8  AInD  ROOTS  79 

Ex.    By  inspection  —  1  is  a  root  of  the  equation 

a;*  +  x3  +  2x2  +  3x  +  l  =  0.  (1) 

Hence  x  +  1  is  a  factor  of  tlie  left-hand  member  of  tlie  equation,  which  may 
accordingly  be  written 

(x  +  l)(x3  +  2x  +  l)  =  0.  (2) 

Additional  roots  of  equation  (1)  may  now  be  found  by  solving  the  equation 
x3  +  2x  +  l  =  0by  methods  given  in  §§  62  and  63. 

It  is  to  be  noted  that  the  solution  of  the  original  equation  has  been  simplified 
by  making  it  depend  upon  the  solution  of  a  depressed  equation,  i.e.  one  of  degree 
lower  than  the  degree  of  the  original  equation. 

41.  By  means  of  the  second  theorem  we  can  form  an  equation 
which  shall  have  any  given  quantities,  r^,  r.,,  •  •  •,  r^  as  roots.  For 
if  r^,  r^,  •  •  •  are  the  roots  of  the  equation,  its  left-hand  member 
must  contain  the  factors  x  —  r^,  x  —  r^,  •  ■  • ,  the  right-hand  mem- 
ber being  zero.    Therefore  the  equation 

(x  —  r^)  (x  —  r^)  ■  '  •  (x  —  rj  =  0 

has  the  required  quantities  as  roots.  Moreover,  this  equation  can 
have  no  other  roots,  since  any  other  value  of  x  will  make  no  fac- 
tor equal  to  zero,  and  hence  the  product  will  not  be  zero.  There- 
fore the  required  equation  is 

(x  —  Vj)  {x  —  r„)---(x  —  r„)  =  0. 

Ex.  1.    Form  the  equation  having  as  roots  2  -f-  3  V  — 1,  2  —  3  V—  1,  —  -J. 
The  required  equation  is 

(a;  _  2  -  3  V^)  (x  -  2  +  3  V^)  (x  +  -J)  =  0, 
or  [(X  -  2)2  +  9]  [3x  +  1]  =  0, 

or  3x3-11x2  4- 35x  + 13  =  0. 

This  method  of  forming  an  equation  suggests  a  method  of  factor- 
ing a  quadratic  expression.  For  if  r^  and  r^  are  the  roots  of  the 
quadratic  equation  ax^+  hx+  e=  0,  then  aaf-i-  hx  +  c  i&  divisible 
by  x  —  r^  and  x  —  r,, ;  and  hence 

ax^  +  bx  +  c  =  a  (x  —  r^)  (x  —  r^). 


80  THE  POLYNOMIAL  OF  THE  .Vth  DEGREE 

Ex.  2.    Factor  6  x^  +  x  -  1. 

The  roots  of  the  equation  6x2  +  x  —  1  =  0  are  _  ^  and  ^. 

,-.  6x2  +  X  -  1  =  6(x  +  ^)  (X  -  ^) 

=  2(x  +  ^).3(x-i) 
=  (2x  +  l)(3x-l). 

Ex.  3.    Factor  4  x^  +  4  x  -  2. 

The  roots  of  the  equation  4x2  +  4x  —  2  =  0  are 

...  4x2  +  4x  -  2  =  4(x  -  ^if^^)  (x  -  :^if^) 

=  (2x  +  1  -  V3)  (2x  +  1  +  V3). 

Ex.  4.    Factor  x2  +  4  x  +  6. 

The  roots  of  the  equation  x2  +  4x  +  6  =  0  are  —  2  ±  V—  2. 

.-.  x2  +  4x  +  6  =  (x  +  2  -  V3^)  (x  +  2  +  V^). 

42.  Number  of  roots  of  an  equation.  The  fundamental  propo- 
sition concerning  the  roots  of  an  equation  is  that  every  equation 
formed  by  placing  a  polynomial  equal  to  zero  has  at  least  one  root. 
The  proof  of  this  proposition,  however,  depends  upon  methods 
too  advanced  to  be  used  here.  We  shall  therefore  assume  it  as 
proved,  and  proceed  to  prove,  as  a  consequence  of  it,  that  every 
equation  of  the  nth  degree  has  n  roots,  and  only  n  roots. 

Let  the  given  equation 

a^af  +  ajOf-^H \.a^_^x+  a^  =  0 

be  denoted  by  f{x)  =0.  (1) 

Since  this  equation  must  have  at  least  one  root,  let  r^  be  that 
root.    Then  f{x)  is  divisible  hy  x~r^  (§  40)  and  therefore 

f{x)  =  {x-r,)f,{x),  (2) 

/j(ic)  being  the  other  factor,  and  necessarily  of  degree  n  —  1. 
Equation  (1)  can  now  be  written 

{^-r,)f,{x)=0,  (3) 

and  any  root  of  f^{x)  =  0  (4) 

is  a  root  oif{x)  =  0  (§  39). 


NUMBER  OF  ROOTS  81 

But  equation  (4)  must  have  at  least  one  root ;  and  if  we  let  r^ 
be  that  root,  and  reason  as  before,  we  may  write 

U(x)  =  (x-T^f,{x),  (5) 

f^ip^  being  of  degree  n  —  1. 

By  substitution  in  (2)  we  shall  have 

f{x)  =  {x-r^(x-T^f^{xy  s  (6) 

After  separating  n  linear  factors  in  this  way,  the  last  quotient 
will  be  (Xq.    Therefore  we  shall  have 

f{x)  =  aj^a  -  r^)  {x-r^)---{x-  rj,  (7) 

the  polynomial  being  expressed  as  the  product  of  n  linear  factors. 
Then  the  equation  f(x)  =  0  may  be  written 

af,{x  —  r^)  {x  —  r^---{x  —  r„)  =  0,  (8) 

whence  it  is  seen  to  have  n  roots  (§  39),  i.e.  r^,  r^,  •  •  •,  r„. 

It  can  have  no  other  roots ;  for  if  we  let  x  have  any  value  other 
than  r^  1\-  ■  -,  ov  r„,  no  factor  of  the  first  member  of  (8)  is  zero, 
and  hence  the  product  in  the  first  member  is  not  equal  to  zero. 
Therefore  the  equation  of  the  nth.  degree  has  n,  and  no  more 
than  n,  roots,  and  the  polynomial  of  the  nth.  degree  can  always 
be  separated  into  n  linear  factors.  In  general,  however,  it  is  not 
possible  to  determine  these  factors  where  n  >  4. 

It  is  to  be  noted  that  the  roots  may  all  be  different,  or  some  of 
them  may  occur  more  than  once.  In  the  latter  case  the  equation 
is  said  to  have  multiple  roots. 

43.  If  now  the  left-hand  member  of  equation  (8)  of  §  42  is 
expanded,  the  equation  appears  in  the  original  form 


a^af  +  a^af'  ^ -j h  a^,_iX  +  «„  =  0, 

and  it  is  evident  that 

(-r,)  +  (-r,)  +  (-r3)-|-...+(-r„)=% 

"■0 

(1) 

and  that                  (-  r,)  (-  r,)  (-  r^)  ■  -  ■  (-  O  =  ^  ' 

(2) 

82    THE  POLYNOMIAL  OF  THE  Ntu   DEGKEE 

Equations  (1)  and  (2)  express  respectively  the  following  theorems: 

1.  The  sum  of  the  roots  of  an  equation  with  their  signs  changed 
is  the  coefficient  of  x^'^  divided  hy  that  of  «". 

2.  The  product  of  the  roots  of  an  equation  vnth  their  signs 
changed  is  the  constant  term,  divided  hy  the  coefficient  of  af. 

Other  theorems  of  this  type  are  given  in  works  on  the  theory 
of  equations,  but  only  these  two  have  been  stated  here,  since  they 
are  of  special  service  in  finding  the  remaining  root  of  an  equation 
after  all  the  others  have  been  determined. 

Ex.  1.  Three  roots  of  the  equation  2 x*  +  Tx^  +  8 a;2  +  2  a;  -  4  =  0  are  -  2, 
_  1  _  V—  1,  and  —  1  +  V—  1.    Find  the  fourth  root. 

The  sum  of  all  the  roots  is  —  ^,  and  the  sum  of  the  three  roots  known  is  —  4. 
Therefore  the  fourth  root  is  —  |  —  (—  4),  or  ^. 

Ex.  2.  Two  roots  of  the  equation  36  x^  —  7  x  +  1  =  0  are  ^  and  —  J.  Find 
the  third  root. 

Tlae  sum  of  the  two  roots  known  is  —  ^,  and  the  sum  of  all  the  roots  is  0, 
since  the  coefficient  of  x^  is  0;  therefore  the  third  root  is  0  —  (—  ^),  or  ^. 

Or  the  product  of  the  roots  known  is  —  ^,  and  the  product  of  all  the  roots 
is  —  3^5  ;  therefore  the  third  root  is  (—  ^^)  -;-  (—  ^),  or  ^. 

44.  Conjugate  complex  roots.  Nothing  was  said  in  §  42  as 
to  the  nature  of  the  roots  r^,  r„,  •  •  • ,  r„.  But  if  the  coefficients 
«o»  «i>  •  •  • »  ^n  ^re  all  real,  and  if  a  +  hi  is  one  of  the  roots,  then 
a  —  hiSs,  also  a  root. 

For  if  a  +  Z>t  is  a  root  of  f{:£)  =  0,  then  f{a  +  hi)  =  0.  When 
f{a  +  hi)  is  expanded  the  terms  can  be  separated  into  two  sets, 
—  those  containing  a  alone  or  involving  only  even  powers  of  hi 
as  a  factor,  and  those  involving  only  odd  powers  of  hi  as  a  factor. 
By  §  12  the  terms  of  the  first  set  are  all  real  and  their  sum  may 
be  denoted  by  A ;  and  the  terms  of  the  second  set  contain  i  to  the 
first  power  as  a  factor,  and  their  sum  may  be  denoted  by  Bi  (B, 
of  course,  being  real).    Then  f{a  +  hi)  =  0  may  be  written 

A  +  Bi  =  0, 

whence  (§  12),  ^  =  0  and  5  =  0. 

If,  in  the  above,  we  replace  hi  by  —  hi,  it  is  evident  that  the 
terms  in  the  first  set  are  not  affected,  as  they  involve  only  even 


GRAPHS  OF  PRODUCTS  83 

powers  of  li  as  a  factor,  and  those  in  the  second  set,  involving 
only  odd  powers  of  hi  as  a  factor,  are  changed  in  algebraic  sign 
only.  Therefore  we  have  f\a  +  (—  hiy]  =  A  —  Bi.  But  we  have 
seen  that  A  =  Q  and  J?  =  0  ;  therefore  f\ct  +  (—  hi)]  =  0.  Since 
f\a  +  (—  11)]=  f  [a  —  hi),  however,  it  follows  that  f{a  —  hi)  =  0, 
and  a  —  &i  is  a  root  of  the  given  equation  f{x)  =  0. 

This  fact  is  usually  stated  by  saying  that  complex  roots  occur 
in  pairs. 

It  follows  that  an  equation  of  even  degree  may  not  have  any 
real  roots,  and  that  an  equation  of  odd  degree  must  have  an  odd 
number  of  real  roots,  and  thus  at  least  one  real  root. 

45.  It  was  proved  in  §  42  that  every  polynomial  is  equivalent 
to  the  product  of  n  linear  factors,  i.e. 

«o(^  -  ^i)  {x-r^---{x-  r„), 

where  r^,  7\,  •  •  • ,  r„  are  the  roots  of  the  corresponding  equation. 
Now  if  any  one  of  these  roots  is  complex,  there  will  be  a  corre- 
sponding conjugate  complex  root.  Let  a  +  hi  and  a  — hi  be  two 
such  roots.  Then  the  corresponding  factors  are  (x  —  a  —  hi)  and 
(x  —  a  +  hi),  which  combine  into  (x—a)^-i-h^,  a  real  quadratic 
factor. 

Therefore  every  polynomial  with  real  coefficients  is  equivalent 
to  the  product  of  real  linear  and  quadratic  factors. 

46.  Graphs  of  products  of  real  linear  and  quadratic  factors. 

1.  All  the  factors  linear  and  none  repeated,  as 
a^{x  -  r,)  {x-r^)---{x-  r„). 

Placing  y  equal  to  this  expression,  we  have 

y  =  a^{x  -  r,)  {x-r,y--{x-  r„). 

It  is  evident  that  the  graph  intersects  the  axis  of  a;  at  ?i  dis- 
tinct points  for  which  x  =■  r^,  x  =  r^,  ■  •  ■ ,  x  =  r^,  and  at  no  other 
points,  as  no  other  values  of  x  make  y  zero.  Now  let  the  quan- 
tities T^,  r^,  ■  •  • ,  r^  be  arranged  in  the  order  of  their  magnitude, 
r^  being  the  least.  Then  if  at  first  x  <  r^,  all  the  factors  are  nega- 
tive ;  and  if  x  changes  so  that  r^<  x  <  r^,  the  first  factor  becomes 
positive  while  all  the  others  remain  negative.    Therefore  y  changes 


84  THE  POLYNOMIAL  OF  THE  Ath  DEGREE 

sign  when  x  changes  from  being  less  than  r^  to  being  greater 
than  rj,  and  the  curve  crosses  the  axis  of  x  at  the  point  x  =  r^. 

Again,  if  x  changes  so  that  at  first  7\<  x  <  i\  and  then 
r^<x<  rg,  the  second  factor  changes  sign  from  minus  to  plus, 
the  others  retaining  their  original  signs.  Hence  y  again  changes 
sign,  and  the  curve  crosses  the  axis  of  x  again  at  the  point  x  =  r^. 

Continuing  in  this  manner,  we  can  show  that  the  curve  crosses 
the  axis  oi  x  n  times  as  it  is  traced  from  left  to  right. 

2.  All  the  factors  linear,  some  being  repeated,  as,  for  example, 

a'o{^-ri){x-r^Y{x-r^)\ 
the  corresponding  equation  being 

y  =  a,{x-  r,)  (x  -  r^Y  {x  -  r^)\ 

If  the  r's  are  arranged  in  ascending  order  of  magnitude,  it  may 
be  proved,  as  in  the  previous  case,  that  the  graph  crosses  the  axis 
of  X  at  the  points  x  =  r^,  and  x  =  r^,  but  not  at  the  point  x  =  r^. 
For  if  at  first  r^<  x  <r^  and  then  r^<x  <  r^,  it  is  seen  that  no 
factor  changes  sign.  But  since  ^  =  0  when  x  =  r„,  the  graph  has 
a  point  on  the  axis  of  x  when  x  =  r^;  in  fact,  it  is  tangent  to  the 
axis  of  X.  And  it  can  be  proved  in  general  that,  if  a  linear  factor 
occurs  an  even  number  of  times,  the  graph  does  not  cross  the  axis 
of  X  at  the  corresponding  point. 

3.  Some  of  the  factors  quadratic,  as,  for  example, 

«o  (^  -  ^i)  (^  -  ^2?  [(^  -  «)'  +  ^']' 
the  corresponding  equation  being 

y  =  a,{x-  r^)  (x  -  r„f  [{x  -  af  +  5=^. 

The  only  new  type  of  factor  is  [x  —  af  +  W,  and  this  is  positive 
for  all  values  of  x.  Hence  there  is  no  new  point  to  be  discussed 
in  regard  to  the  intersection  of  the  graph  with  the  axis  of  x. 

In  general,  the  graph  has  as  many  points  on  the  axis  of  x  as 
the  polynomial  has  different  linear  factors;  it  does  not  cross  the  axis 
at  any  point  corresponding  to  a  factor  occurring  an  even  number 
of  times ;  and  it  crosses  the  axis  of  x  at  any  point  corresponding 
to  a  factor  occurring  an  odd  number  of  times. 


EXAMPLES  OF  GRAPHS 


85 


Ex.1.    2/  =  .5(x  +  2)(x+.5)(a;-2). 

1.  If  X  =  —  2  or  —  .  5  or  2,  y  =  0, 
and  there  are  three  points  of  the 
curve  on  the  axis  of  x. 

2.  If  X  <  —  2,  all  three  factors  are 
negative  ;  therefore  2/  <  0,  and  the 
corresponding  part  of  the  curve  lies 
below  the  axis  of  x.  If  —  2  <  x  <  — .  6, 
the  first  factor  is  positive  and  the 
other  two  are  negative ;  therefore 
2/  >  0,  and  the  corresponding  part  of 
the  curve  lies  above  the  axis  of  x. 
If  —  .5  <  X  <  2,  the  first  two  factors 
are  positive  and  the  third  is  nega- 
tive ;  therefore  y  <0,  and  the  corre- 
sponding part  of  the  curve  lies 
below  the  axis  of  x.  Finally,  if 
X  >  2,  all  the  factors  are  positive ; 
therefore  y>0,  and  the  correspond- 
ing part  of  the  curve  lies  above  the 
axis  of  X. 

3.  Assuming  values  of  x  and 
finding  the  corresponding  values 
of  2/,  we  plot  the  curve,  as  repre- 
sented in  fig.  41. 


Fig.  41 


Fig.  42 


Ex.2.    y  =  .5(x  +  2.5)(x-l)2. 

1.  If  X  =  —  2.5  or  1,  2/  =  0,  and  there 
are  two  points  of  the  curve  on  the 
axis  of  X. 

2.  If  X  <  —  2.5,  the  first  factor  is 
negative  and  the  second  factor  is  posi- 
tive ;  therefore  y  <0,  and  the  corre- 
sponding part  of  the  curve  lies  below 
the   axis   of  x.     If    —  2.5<x<l,   both 

■X  f Victors  are  positive ;  therefore  y>0, 
and  the  corresponding  part  of  the  curve 
lies  above  the  axis  of  x.  Finally,  if 
«  >  1,  we  have  the  same  result  as  when 
—  2.6<x<l,  and  the  curve  does  not 
cross  the  axis  of  x  at  the  point  x  =  1, 
but  is  tangent  to  it. 

3.  Assuming  values  of  x,  and  finding 
the  corresponding  values  of  y,  we  plot 
the  curve  as  represented  in  fig.  42. 


86 


THE  POLYNOMIAL  OF  THE  Nth   DEGREE 


Ex.  3.    2/  =  .5(x  +  3) 

(x2- 2.5  a; +  3.5). 

■  1.  If  X  =  —  3,  y  =  0,  and  this 
curve  has  but  one  point  on  the 
axis  of  X. 

2.  If  X  <  —  3,  the  first  factor  is 
negative  and  the  second  factor  is 
positive,  as  it  always  is,  since  it  is 
equivalent  to  (x  —  1.25)2  +  1.9375; 
therefore  y  <0,  and  the  corre- 
sponding part  of  the  curve  is 
below  the  axis  of  x.  If  x  >  —  3, 
the  first  factor  is  positive ;  there- 
fore 2/  >  0,  and  the  corresponding 
part  of  tlie  curve  is  above  the 
axis  of  X. 

3.  Assuming  values  of  x,  and 
finding  the  corresponding  values 
of  y,  we  plot  the  curve  as  repre- 
sented in  fig.  43. 

47.  Location  of  roots. 
From  the  work  of  the  last 
article  it  is  evident  that  the 
real  roots  of  the  equation  f(x)=0  determine  points  on  the  axis 
of  X  at  which  the  graph  of  f{x)  crosses  or  touches  that  axis. 
Moreover,  if  x^  and  x^  (x^  <  x^)  are  two  values  of  x,  such  that 
/(Xj)  and  f{x^)  are  of  opposite  algebraic  sign,  the  graph  is  on  one 
side  of  the  axis  when  x  =  x^,  and  on  the  other  side  when  x  =  x^. 
Therefore  (§56)  it  must  have  crossed  the  axis  an  odd  number  of 
times  between  the  points  x  —  x^  and  x  =  x^.  Of  course  it  may- 
have  touched  the  axis  at  any  number  of  intermediate  points. 
Since  a  point  of  crossing  corresponds  to  an  odd  number  of  roots 
of  an  equation,  and  a  point  of  touching  corresponds  to  an  even 
number  of  roots,  it  follows  that  the  equation  f(x)  =  0  has  an  odd 
number  of  real  roots  between  x^^  and  x^. 

The  above  gives  a  ready  means  of  locating  the  real  roots  of 
an  equation  in  the  form  f(x)  =  0,  for  we  have  only  to  find  two 
values  of  x,  as  x^  and  x^,  for  which  f{x)  has  different  signs.  We 
then  know  that  the  equation  has  an  odd  number  of  real  roots 
between  these  values,  and  the  nearer  together  x^^  and  x^,  the  more 


Fig.  43 


DESCARTES'  RULE  OF  SIGNS  87 

nearly  do  we  know  the  values  of  the  intermediate  roots.  In  locat- 
ing the  roots  in  this  manner  it  is  not  necessary  to  construct  the 
corresponding  graph,  though  it  may  be  helpful. 

48.  Descartes'  rule  of  signs.  When  in  a  polynomial  a  term 
with  one  sign  is  immediately  followed  by  one  with  the  opposite 
sign,  there  is  said  to  be  a  variation  of  sign.  For  example,  in  the 
polynomial  3  aj*  +  2  a;^  —  3  ic^  +  ic  —  2  there  are  three  variations. 

The  variations  of  sign  in  the  left-hand  member  of  an  equation 
are  often  of  value  in  locating  the  real  roots  of  the  equation,  for 
the  number  of  positive  roots  of  the  equation  f  (x)  =  0  cannot  exceed 
the  number  of  variations  of  sign  in  its  left-hand  member.  This 
rule  is  known  as  Descartes'  rule  of  signs. 

For  example,  the  equation  3x^-i-2x^  —  3x^  +  x— 2  =  0  cannot 
have  more  than  three  positive  roots,  as  there  are  three  variations 
of  sign  in  its  left-hand  member. 

To  determine  the  greatest  possible  number  of  negative  roots, 
replace  ic  by  —  x'.  The  roots  of  the  resulting  equation  will  be 
those  of  the  original  equation  with  their  signs  changed.  Accord- 
ingly the  original  equation  can  have  no  more  negative  roots  than 
this  new  equation  has  positive  roots. 

If,  in  the  equation  3  x*  -\-  2  x^  —  3  x'^  +  x  —  2  =  0,  x  is  replaced 
by  —  x',  tlie  new  equation  is  3  x'*  —  2  x'^  —  3  x'^  —  x'  —  2  =  0.  As 
this  equation  cannot  have  more  than  one  positive  root,  the  original 
equation  cannot  have  more  than  one  negative  root. 

Sometimes,  l)y  Descartes'  rule,  we  can  prove  that  an  equation 
has  imaginary  roots.  For  example,  the  equation  3  a;^  -|-  «^  -f-  2  =  0 
can  have  no  positive  root,  and  not  more  than  one  negative  root. 
Being  of  odd  degree,  it  has  at  least  one  real  root  (§  44);  therefore 
it  has  one  negative  root  and  two  imaginary  roots. 

In  order  to  prove  Descartes'  rule  we  will  first  prove  that  if  any 
polynomial  f(x)  is  multiplied  by  x  —  r,  ivhere  r  is  a  positive  quan- 
tity, the  product  has  at  least  one  more  variation  than  has  f{x). 

Assuming  the  first  term  of  f{x)  to  l)e  positive,  we  will  inclose 
all  the  terms  preceding  the  first  minus  §ign  in  a  parenthesis.  In 
a  second  parenthesis  we  will  inclose  all  the  terms  with  a  minus 
sign  before  a  positive  sign  occurs  again,  and  so  on.  Suppose, 
then,  that  the  first  minus  sign  appears  in  the  term  containing 


88  THE  POLYNOMIAL  OF  THE  Nth  DEGREE 

x""*,  the  next  plus  sign  occurs  in  the  term  containing  af-\  etc., 
and  that  all  the  terms  after  that  containing  x"""  have  the  same 
sign  as  that  term.    We  can  now  write 

—  {a^^af-^  H \-  a,_iaf -'+') 

±K^"'"  + •••  +  ««)>  (1) 

where  all  the  terms  within  each  parenthesis  are  of  the  same  sign, 
i.e.  plus.    Therefore  each  parenthesis  marks  a  variation. 

To  multiply  f(x)  hj  x  —  r  we  shall  multiply  first  by  x,  then  by 
—  r,  and  add  the  partial  products. 

The  result  is  an  equation  of  the  following  form :  ' 

+  (&^-'  +  i±...) 

±(&„af-"'  +  ^±...)HFa„r,  (2) 

where  h^.  =  a^  +  ra^._i,  hi  =  ai  +  rai_^,  etc.,  and  accordingly  are 
positive. 

The  signs  before  each  parenthesis  of  (2)  are  the  same  as  in  (1), 
but  the  signs  within  the  parenthesis  are  not  necessarily  all  plus. 
But  however  the  signs  may  occur  within  any  parenthesis,  there 
is  at  least  one  variation  between  tlie  first  term  of  one  parenthesis 
and  the  first  term  of  the  following  parenthesis.  Hence,  if  we  con- 
sider the  parentheses  only,  the  number  of  variations  in  the  prod- 
uct is  not  less  than  the  number  of  variations  in  f{x). 

But,  in  addition,  we  have  the  last  term  of  the  product,  i.e.  :f  a„r, 
the  sign  of  which  differs  from  the  sign  of  the  first  term  in  the 
last  parenthesis.  Hence  there  is  at  least  one  more  variation  in 
{x  —  r)  f{x)  than  in  f{x),  as  we  set  out  to  prove. 

Now  the  equation  having  the  roots  r^,  rg,  •  •  • ,  r„  is  (§  41) 

{X-  T^)  {X-  r^)  ■  ■  ■  {X  -  t;)=  {). 

In  expanding  the  left-hand  member  every  time  we  multiply 
by  a  factor  corresponding  to  a  positive  root,  we  add  at  least  one 
variation  of  sign.  Hence  the  number  of  positive  roots  cannot 
exceed  the  number  of  variations,  as  stated  in  Descartes'  rule. 


EATIONAL  BOOTS  89 

49.  Rational  roots.  The  real  roots  of  any  equation  are  either 
rational  or  irrational  (§  10),  and  the  rational  roots  must  be 
either  integral  or  fractional.  We  will  now  derive  methods  of 
finding  the  rational  roots,  beginning  with  the  integral  roots. 

An  easy  method  of  determining  the  integral  roots  depends  upon 
the  following  theorem :  If  the  equation  is  written  in  the  forni 

a^a^  +  a^a""-^  H h  «„_!«;  + a„  =  0,  (1) 

ivhere  all  the  coefficients  are  integers,  any  integral  root  r  must  he 
a  factor  of  a„. 

It  has  been  proved  in  §  40  that  the  left-hand  member  of  (1)  is 
divisible  hy  x  —  r.  Since  the  coefficient  of  x  is  unity,  and  all  the 
coefficients  in  the  dividend  are  integers,  all  the  coefficients  in  the 
quotient  are  integers.  But  the  last  coefficient  in  the  quotient 
multiplied  by  r  must  be  «^„,  since  there  is  no  remainder.  Hence 
the  theorem  is  proved. 

Accordingly,  to  find  the  integral  roots  of  any  equation  with 
integral  coefficients,  we  have  merely  to  try  the  integral  factors  of 
a„.  When  an  integral  root  has  been  found,  we  depress  the  degree 
of  the  equation  as  in  §  40,  and  apply  the  process  to  the  new 
equation.  In  this  way  all  the  integral  roots  may  be  found.  In 
case  no  integral  factor  of  a„  proves  to  be  a  root,  it  follows  that 
the  equation  can  have  no  integral  root. 

Ex.  Find  the  integral  roots  of  the  equation 

4 X*  -  4 x3  -  26x2  +  X  +  6  =  0. 

The  integral  roots  of  this  equation  must  be  factors  of  6,  so  that  we  have  to 
tiy  ±  1,  ±  2,  ±  3,  ±  6.  By  trial  it  is  found  that  —  2  is  a  root,  and  the  degree 
of  the  equation  is  depressed  by  dividing  the  left-hand  member  by  x  -1-  2,  the 
depressed  equation  being  4  x^  —  12  x^  —  x  -H  3  =  0.  The  only  possible  values  of 
integral  roots  of  this  equation  are  db  1,  ±  3,  and  3  is  found  to  be  a  root.  Dividing 
the  left-hand  member  by  x  —  3,  we  have,  as  the  depressed  equation,  4x2—  1  =  0, 
the  roots  of  which  are  ±  ^. 

Therefore  the  roots  of  the  original  equation  are  —  2,  3,  ±  J. 

While  all  the  integral  roots  of  an  equation  may  be  found  by 
tliis  method,  it  is  evident  that  it  fails  for  fractional  roots,  as  there 
is  no  way  of  determining  what  fractions  ought  to  be  tried.  This 
difficulty  is  obviated  by  the  two  theorems  in  the  next  article. 


90    THE  POLYNOMIAL  OF  THE  3th  DEGREE 

50.  If  a^  is  unity  and  all  the  other  coejgHcients  are  integers, 
the  equation  cannot  have  a  rational  fraction  in  its  lowest  terms 
as  a  root. 

Let  the  equation  be 

af  +  a^af-'^+a^af-^-\ ha„_^x  +  a„=0, 

p 
and,  if  possible,  let  the  rational  fraction  - ,  wliich  is  in  its  lowest 

terms,  be  a  root.    Then 

(f)"-'(f)""-'(f)""--»-(f)---- 

Multiply  through  by  q"'^,  and  transpose  to  the  second  member 
all  terms  but  the  first.    Then 

^  =  -  a^p"-'-  aj)''-\ "n-ii??""'  -  «„2""'- 

By  hypothesis  p  and  q  have  no  common  factor,  and  therefore  — 

is  a  rational  fraction  in  its  lowest  terms,  while  the  right-hand 
member  of  the  equation  is  an  integral  expression.    But  two  such 

p 
expressions  cannot  be  equal,  and  hence  — ,  the  rational  fraction 

in  its  lowest  terms,  cannot  be  a  root  of  the  equation. 
Moreover,  every  equation  in  the  form 

ttoX"  +  ttjO^-'  +  a^cff"--  -\ h  a^_^x  +  a„  =  0, 

in  which  a^  is  not  unity,  can  he  tra-nsformed  into  an  equation  with 
integral  coefficients  in  tvhich  the  coefficient  of  the  highest  power  of 
the  unknow7i  quantity  shall  he  unity. 
For,  dividing  tlirough  by  «„,  we  have 

a.  a„  a    ,         a 

ic"+-iiC»-i  +  -^af'-2-|-...-f--^^a;-}--  =  0.  (1) 


If  «  is  a  root  of  this  equation,  let  a?  =  —  >  7n.  being  an  integer,  and 
substitute  in  (1).    Then 

w"      ffoWi  a.m'^  «„   m      a,,  ^  ' 


RATIONAL  ROOTS  91 

Multiplying  (2)  by  wi",  we  have 


x"'-\-[-^m  x'"-'  +  {-^m^]x'"-'  + 


In-l,l      2        2l^./n-2 


We  can  now  determine  m  by  inspection  in  such,  a  way  that 
all  the  coefficients  of  (3)  shall  be  integers.  The  roots  of  this  new 
equation  are  m  times  the  roots  of  the  original  equation. 


Ex.  Transform  equation  12x3  +  16a;2  _  6x  —  3  =  0  to  an  equation  having 
integral  coeflBcients,  the  coefficient  of  the  highest  power  of  x  being  unity. 
Dividing  by  12,  we  have 

Multiplying  the  roots  of  this  equation  by  an  integer  m,  we  insert  in  each 
term  a  power  of  to  such  that  the  sum  of  its  exponent  and  that  of  x'  shall  be 
equal  to  the  degree  of  the  equation,  thus  obtaining 

X'8  +  (J  to)  X'2  -  (^\  to2)  X'  -  (4  7n,8)  =  0. 

For  ^  m  to  be  an  integer,  to  must  equal  3  k  where  k  is  an  integer.  Then  ^'^  m2 
becomes  j''^  (9  k^),  and  this  is  an  integer  only  when  k  =  21;  i.e.  in  =  61,  I  being 
an  integer.  Finally,  ^  to^,  or  ^  (6  Z)^,  is  an  integer  if  1  =  1,  the  least  value  of  to 
being  the  one  desired. 

Therefore  we  let  to  =  6,  and  our  required  equation  is 

x'3  +  8x'2  -  15x'  -  64  =  0, 

the  roots  of  which  are  six  times  the  roots  of  the  original  equation. 

The  roots  of  this  equation  are  found  by  the  method  of  §  49  to  be  —  2,  3, 
and  —  9.    Hence  the  roots  of  the  original  equation  are  —  ^,  ^,  and  —  ^. 

We  are  thus  in  a  position  to  determine  the  rational  fractional 
roots  of  any  equation  with  rational  coefficients. 

51.  We  now  see  that  to  find  all  the  rational  roots  of  any  equa- 
tion, we  first  find  all  its  integral  roots  and  then  all  its  fractional 
roots,  as  indicated  in  the  following  example. 


92    THE  POLYNOMIAL  OF  THE  .Vth  DEGREE 

Ex.    Find  all  the  rational  roots  of  the  equation 

2x*-5x3_2a:2-7x  +  30  =  0.  (1) 

By  Descartes'  rule  of  signs  this  equation  cannot  have  more  than  two  posi- 
tive roots,  and  not  more  than  two  negative  roots.  If  any  of  the  roots  are  inte- 
gral, they  will  be  among  the  factors  of  30,  i.e.  ±  1,  ±  2,  ±  3,  ±  5,  ±  6,  ±  10, 
±  16,  ±  30.    By  trial  we  find  +  2  to  be  a  root,  and  the  depressed  equation  is 

2x8 -x2-4x- 15  =  0.  (2) 

By  trial  we  find  that  this  new  equation  has  no  integral  roots,  no  factor  of  15 
being  a  root.    Accordingly  we  proceed  to  find  fractional  roots. 

Dividing  equation  (2)  through  by  2  and  then  multiplying  the  rpots  by  m,  we 

have  x'3  -  (^  m) x'2  -  (2  m^)  z'  -  (V-  rnS)  =  0.  (3) 

To  make  the  coefficients  of  (3)  integral  we  take  wi  =  2,  and  the  equation  becomes 

a;'8_x'2_8x'-60  =  0.  (4) 

By  trial  we  find  an  integral  root  of  this  equation  to  be  5,  and  the  depressed 

equation  is  „      .         ,„      „ 

^  x2-|-4x  +  12  =  0,  (5) 

the  roots  of  which  are  —  2  ±  2  V—  2. 

Therefore  the  three  roots  of    the   transformed    equation  (4)   are    5  and 

—  2  ±  2  V—  2,  and  the  roots  of  the  first  depressed  equation  (2)  are  ^  and 

—  1  ±  V—  2,  so  that  the  roots  of  the  given  equation  are  2,  ^,  and  —  1  ±  V—  2. 
It  is  to  be  noted  that  in  this  example,  after  having  found  all  the  rational 

roots,  we  were  able  to  find  the  remaining  roots  also,  since  the  last  depressed 
equation  was  of  no  higher  degree  than  the  second. 

52.  Irrational  roots.  It  should  be  borne  in  mind  that  rational 
roots  occur  only  for  special  values  or  systems  of  values  of  the 
coefficients.  Hence,  after  removing  the  rational  roots,  if  any,  by 
the  previous  methods,  we  have,  in  general,  to  determine  irrational 
roots  in  order  to  have  all  the  real  roots  of  the  equation.  But 
from  the  definition  of  an  irrational  quantity  (§  10)  it  is  evident 
that  we  cannot  find  an  irrational  root  exactly.  We  may,  however, 
find  an  approximate  value  to  any  required  degree  of  accuracy. 
There  are  various  methods  of  approximation,  one  of  which  imme- 
diately follows.     A  more  rapid  method  is  given  in  §  63.* 

*  A  method  of  solving  algebraic  equations,  known  as  Horner's  method,  is  found 
in  most  treatises  on  the  theory  of  equations.  It  is  convenient  in  arrangement  of 
work  and  speedy  in  the  hands  of  an  expert.  It  may  therefore  be  recommended  to 
one  who  has  often  to  solve  equations.  On  the  other  hand,  the  methods  of  §§  52,  6.3  of 
this  book  have  two  advantages.  They  may  be  applied  to  other  than  algebraic  equa- 
tions (see  §  162),  and  depend  upon  principles  which,  if  once  mastered,  are  not  easily 
forgotten. 


IRRATIONAL  ROOTS 


93 


Fig.  44 


Let  the  given  equation  'bef(x)=0,  and  the  graph  of  the  left- 
hand  member  be  as  in  fig.  44,  where  OM^  =  x^  and  OM^  =  x^. 
Then  M^F^  =f{x^)  and  M^P^=f{x^,  and  since  f{x^  and  f{x^  are 
of  opposite  sign,  the  curve  crosses  the  axis  of  x  between  M^  and 
M„,  and  there  is  at  least  one 
real  root  of  f{x)=Q  between 
x^  and  x^  (§  47). 

Not  only  does  the  curve  cross 
the  axis  of  x  at  some  point  be- 
tween Jfj  and  M^,  but  it  is 
evident  from  fig.  44  that  the 
straight  line  P^P^  also  intersects 
the  axis  of  x  at  some  point 
between  M^  and  M^,  as  M^.  If 
the  points  M^  and  M^  are  near 
together,  i.e.  if  x^  and  x^  differ 
only  by  a  small  amount,  the  curve  in  most  cases  differs  only  slightly 
from  the  straight  line  P^P^.  Hence,  if  we  replace  the  curve  by 
the  straight  line,  the  abscissa  of  the  point  at  which  P^P^  intersects 
the  axis  of  x  will  be  approximately  the  root  of  the  equation. 

If  OJ/3  is  denoted  by  x^,  it  is  evident  (fig.  44)  that  there  is  a 
root  of  f(x)=0  between  x^  and  x^,  a  smaller  interval  than  that 
between  x^  and  x^,  in  which  the  root  was  first  located. 

If,  however,  the  graph  oi  f{x)  had 
been  as  in  fig.  45,  the  root  would 
have  been  between  x^  and  x^,  an 
interval  smaller,  of  course,  than  that 
between  x^  and  x^. 

If  f{x^  has  the  same  sign  as  f{x^, 
we  have  the  first  case  (fig.  44) ;  and 
if  f{x^  has  the  same  sign  as  f{x.^, 
we  have  the  second  case  (fig.  45). 
In  the  first  case,  repeating  the  proc- 
ess, using  iCg  in  place  of  x^,  we  can 
find  an  x^  between  which  and  x^ 
the  root  must  lie ;  and  in  the  second  case,  using  x^  in  place  of  x^, 
we  can  find  an  x^  between  which  and  j\  the  root  must  he. 


94  THE  POLYNOMIAL  OF  THE  Nth  DEGREE 

Moreover,  it  is  evident  that  the  successive  values  of  x,  ie.  x^, 
«4)  ^6>  •  •  •>  found  in  this  way  are  each  nearer  to  the  true  value  of 
the  root  of  f(x)=0  than  the  one  preceding. 

Ex.    Find  the  root  of  the  equation  x3  +  2x  —  17  =  0  between  2  and  3. 
Here  Xi  =  2  and  X2  =  3  ;  al8o/(2)  =  -  5  and/(3)  =  16.    The  equation  of  the 
straight  line  determined  by  the  points  (2,  -  5)  and  (3,  16)  is  (§  29) 

,       -  5  -  16  , 
y  +  5=     ^_3     (X  -  2). 

Its  intercept  on  OX,  found  by  letting  y  =  0,  is  2.2  +,  and  /(2.2)  =  —  1.952. 

Since /(2.2)  has  the  same  sign  as/(2),  the  second  straight  line  is  determined 
by  the  points  (2.2,  -  1.952)  and  (3,  16).  Its  intercept  on  OX  is  2.28  +,and 
/(2.28)  =  - 0.587648. 

Since  /(2.28)  and  /(2.2)  have  the  same  sign,  the  third  straight  line  is 
determined  by  the  points  (2.28,  -0.587648)  and  (3,  16).  Its  intercept  on 
OX  is  2.3+,  and  /(2.3)  =  —  0.233.  The  fourth  straight  line  is  determined 
by  the  points  (2.3,  -0.233)  and  (3,16).  Its  intercept  on  OX  is  2.31  +,  and 
/(2.31)  =  -  0.053609.  The  fifth  straight  line  is  determined  by  (2.31,  -0.053609) 
and  (3,  16).    Its  intercept  on  OX  is  2.312. 

Hence  the  irrational  root  of  x^  +  2  x  —  17  =  0,  accurate  to  two  places  of 
decimals,  is  2.31. 

By  continuing  this  process  we  can  find  any  desired  number  of  decimal  places 
of  the  root.  It  is  to  be  noted  that  we  are  obliged  to  find  one  more  decimal  place 
than  the  number  of  decimal  places  to  which  the  root  is  to  be  accurate.  The 
approximation  is  more  rapid  if  the  first  decimal  place  is  found  by  the  method 
of  §  47. 

PROBLEMS 

Plot  the  graphs  of  the  following  quadratic  expressions,  in  each  case  locating 
the  vertex  of  the  graph  and  determining  the  nature  of  the  roots  of  the  corre- 
sponding equation : 

1.  2x2  +  3x-2.  4.   -3x2  +  5x. 

2.  9x2-3x-2.  -  5.   -9x2  +  12x-7. 

3.  4x2  +  4x  +  3.  6.  4x2-4x-l. 

7.  For  what  values  of  a  are  the  roots ofax2  +  3x  +  7  =  0  equal  ?  What  are 
the  roots  ? 

8.  Prove  that  the  roots  of  (ex  +  —J  -  8 ax  =  0  are  equal  for  all  values 
of  a  and  c,  and  find  them.  ^  ^ 

9.  Prove  that  there  is  no  real  value  of  m  for  which  the  roots  of 
x2  +  (mx  +  3)2  -  16  =  0  are  equal. 


PKOBLEMS  95 

For  what  values  of  k  are  the  roots  of  the  following  quadratic  equations  (1) 
equal  ?  (2)  real  and  unequal  ?  (3)  imaginary  ? 

10.  2x2  + 3a; +  2  =  A;.  11.  x2  +  (2  -  A;)x  +  1  =  0. 

12.  {k  +  l)x2  +  (A;  -  l)x  +  (^  +  1)  =  0. 

Plot  the  graphs  of  the  following  polynomials : 

13.  x3  -  ax.    (a  >  0.)  19.  x^  -  12  x  +  3. 

14.  x3  -  4x2  +  X  +  1.  20.  2x*  +  x3  -  4x2  -  lOx  -  4. 

15.  x8- 3x2  +  1.  21.  4x*  +  12x3  +  7x2-28x-6. 

16.  x3  +  x2  +  2x  +  5.  '  22.  3x*  -  10x3  -  5x2  ^  2x. 

17.  x3  -  x2  +  X  -  4.  23.  X*  +  6 x8  +  10x2. 

18.  x3  +  6x-6.  24.  2x5  +  2x*-7x3-8x2-4x. 

Find  all  the  roots  of  the  following  equations : 

25.  8x3  =  27.  28.  5x6  + 27x2  =  2x«-64x*. 

26.  8x6  -  63x8  -  8  =  0.  29.  {2x  -  a)*  -  (3x  +  a)*  =  0. 

27.  x6-5x3  +  12x  =  2x3  +  3x.  30.  x*- 2(a2+l)x2  + (a2 -1)2  =  0. 

Form  the  equations  having  the  following  values  for  their  roots  : 
31.0,2,3.  32.  a  +  Vb,a-Vb,  -a. 

33.  0,  0,  2 a  ±  6,  ±  V2b. 

34.  Form  a  quadratic  equation  with  real  coefficients  having  2  +  3i  for  one 
of  its  roots. 

Factor  the  following  quadratic  expressions : 

35.  4x2  + 8x- 7.  38.  x2  +  2ax- a  + a2. 

36.  4x2  +  12  X  +  11,  39,  ^2x2  +  2  a6x  -  a. 

37.  4  a2x2  +  2  ax  +  1.  40.  a2x2  +  2abx  +  b  +  b'l 

If  ri  and  r^  are  the  roots  of  the  equation  x2  +  px  +  g  =  0,  find  the  values  of 
the  following  expressions  in  terms  of  p  and  q  without  solving  the  equation  : 

41.  r2  +  r|.      42.  rf  +  r.i'.      43.1  +  1.        44.  1  +  i.       45.^  +  ^. 

If  ^ii  ''25  ra  are  the  roots  of  the  equation  x^  +  _px2  +  gx  +  r  =  0,  find  the  val- 
ues of  the  following  expressions  in  terms  of  the  coefficients  without  solving  the 
equation :  ,   .  ' 

46.  (rl  +  r|  +  r.2)  +  2  (nrz  +  rgrs  +  nn)  +  3  rirgrs- 

47.  r{r2r3  +  rhaTi  +  rhir^.  48. h 1 

■^  nrz      r2r3      rsn 

49.  Show  that  if  a  +  VS  is  a  root  of  an  equation  with  rational  coefficients, 
then  a  —  Vft  is  also  a  root. 


96    THE  POLYNOMIAL  OF  THE  Nth   DEGREE 

Plot  the  graphs  of  the  following  expressions,  and  find  all  the  roots  of  the 
corresponding  equations : 

50.  (X  +  1) (a;  -  2)  (z  -  4).  56.  (2z  +  5)(x2  +  2x  +  3). 

51.  (x-2)(x-4)(2x  +  3).  57.  (x-5)(2x2  +  3x  +  2). 

52.  (X  -  4)  (2x  +  1)  (3x  +  5).  58.  (x  +  2)  (x  -  3)  (x  -  2)2. 

53.  (X  +  3)  (X  -  1)2.  59.  (X  -  2)  (X  +  2)  (x2  +  2). 

54.  (2x  -  l)(x  -  3)2.  60.  (X  -  2)2(2x2  +  2x  +  1). 

55.  (x-2)(2x  +  3)2.  61.  (x  +  l)(2x-l)(3x2  +  2x  +  3). 

Find  all  the  roots  of  the  following  equations : 

62.  x»  -  4x2  -  2x  +  6  =  0.  67.  8x3  -  28x2  +  30x  -  9  =  0. 

63.  x8  -  3 x2  +  4  =  0.  68.  12x8  -  44 x2  +  5x  +  7  =  0. 

64.  3x3  _  7x2  -  8x  +  20  =  0.  69.  3x3  +  10x2  +  iqx  -  12  =  0. 

65.  4x3-8x2-35x4-75  =  0.  70.  3x3  +  10x2  +  2x  -  8  =  0. 

66.  x8  +  4x2  +  4x  + 3  =  0.  71.  4x<  +  8x3  +  3x2  -  2x  -  1  =  0. 

72.  6x*  -  11x8  -  37x2 +  36X  + 36  =  0. 

73.  3x<  -  17x3  4- 41x2  -  53x  +  30  =  0. 

74.  2x*  -  9x3  -  9x2  +  o7x  -  20  =  0. 

75.  18x<  -  27x3  +  10x2  +  I2x  -  8  =  0. 

76.  16x*  +  16x8  -  72x2  -  20x  +  25  =  0. 

77.  x8  -  2x<  -  4x3  -  4x2  +  15x  +  18  =  0. 

78.  4x5  +  12x*  +  11x3  +  5x2  -  3x  -  2  =  0. 

79.  12x5  +  44x*  -  55x3  -  95x2  +  63x  -  9  =  0. 

80.  2x6  -  6x*  -  13x3  +  13x2  +  5x  -  2  =  0. 

Determine  by  Descartes'  rule  of  signs  the  nature  of  the  roots  of  the  follow- 
ing equations : 

81.  x3  +  6x-7  =  0.  84.  3x*  +  4x8  +  4x  +  3  =  0. 

82.  x3  +  2x  +  3  =  0.  85.  X*  +  x2  -  X  -  6  =  0. 

83.  x8  +  2x2  +  5  =  0.  86.  X*  -  4x2  +  1  =  0. 

Find  the  real  roots  of  the  following  equations,  accurate  to  two  decimal  places  : 

87.  a^  +  3x  -  7  ^  0.  89.  X*  -  12x  +  7  =  0.. 

88.  i8  +  X  +  5  =  0.  90.  X*  -  3x3  +  3  =  0. 

91.  x3-x2-6x  +  l  =  0. 


f 


I 


CHAPTEK  V 

THE  DERIVATIVE  OF  A  POLYNOMIAL 

53.  Limits.  A  variable  is  said  to  approach  a  constant  as  a 
limit,  when,  under  the  law  ivhich  governs  the  change  of  value  of 
the  variable,  the  difference  between  the  variable  and  the  constant 
becomes  and  remains  less  than  any  quantity  which  can  be  narned, 
no  matter  how  small. 

If  the  variable  is  independent,  it  may  be  made  to  approach  a 
limit  by  assigning  to  it  arbitrarily  a  succession  of  values  follow- 
ing some  known  law.    Thus,  if  x  is  given  in  succession  the  values 

2"  — 1 

■^1  ~"   2"'     "^2  ~"  1»    "^3  —  t'  »    "^n  ~        On 

and  so  on  indefinitely,  it  approaches  1  as  a  limit.  For  we  may 
make  x  differ  from  1  by  as  little  as  we  please  by  taking  n  suffi- 
ciently great ;  and  for  all  larger  ^  ^ 

values  of  n  the  difference  be-   ? | f T  ¥  / 

tween  x  and  1  is  still  smaller.  *  234 

This    may    be    made    evident 

graphically  by  marking  off  on  a  number  scale  the  successive  values 
of  X  (fig.  46),  when  it  will  be  seen  that  the  difference  between  x 
and  1  soon  becomes  and  remains  too  minute  to  be  represented. 
Similarly,  if  we  assign  to  x  the  succession  of  values 

«i  =  J,  ^2=  —  ^,  «^3  =  -4,  ^4  =  ~  5'  ■  ■  ■ '  ^n  —  \~^)       n  +  \  ' 
X  approaches  0  as  a  limit  (fig.  47). 


~s~7  0         s  e       i  2  f 

-t— ) 1 H 1 + 1 

Fig.  47 


If  the  variable  is  not  independent  but  is  a  function  of  x,  the 
values  which  it  assumes  as  it  approaches  a  limit  depend  upon 

97 


98 


THE  DERIVATIVE  OF  A  POLYNOMIAL 


For  example,  let  y  =f{x), 


the  values  arbitrarily  assigned  to  x. 
and  let  x  be  given  a  set  of  values 

approaching  a  limit  a.    Let  the  corresponding  values  of  y  be 

Vv     2/2'     2/3.     2/4'     •'•>     Vny     •••• 
Tlien  if  there  exists  a  number  A,  such  that  the  difference  between 
y  and  A  becomes  and  remains  less  than  any  assigned  quantity,  y 

is  said  to  approach  ^  as  a  limit 
as  X  approaches  a  in  the  man- 
ner indicated.    This  may  be  seen 
graphically  in  fig.  48,  where  the 
values  of   x  approacliing    a   are 
seen  on  the  axis  of  abscissas  apd 
the  values  of   y  approaching  A 
are  seen  on  the  axis  of  ordinates. 
The  curve  of  the  function  is  con- 
tinually nearer  to  the  line  y  =A. 
In  the  most  common  cases,  the 
limit  of  the  function  depends  only 
upon   the  limit  a   of.  the  inde- 
pendent variable  and  not  upon  the  particular  succession  of  values 
that  X  assumes  in  approaching  a.    This  is  clearly  the  case  if  the 
graph  of  the  function  is  as  drawn  in  fig.  48. 

Ex.  1.    Consider  the  function 

x2  +  3  X  -  4 

y  = z ' 

X  —  1 

and  let  x  approach  1  by  passing  through  the  succession  of  values 

x  =  l.l,     x  =  1.01,     x  =  1.001,     x  =  1.0001,     ••.. 

Then  y  takes  in  succession  the  values 

2/ =  5.1,     77=5.01,     y  =  5-001,     2/ =  5.0001. 

It  appears  as  if  y  were  approaching  the  limit  5.  To  verify  this,  we  place  x  =  1  +  A, 
where  h  is  not  zero.  By  substituting  and  dividing  by  h  we  find  2/  =  6  +  A. 
From  this  it  appears  that  y  can  be  made  as  near  5  as  we  please  by  taking  h 
sufficiently  small,  and  that  for  smaller  values  of  h,  y  is  still  nearer  5.  Hence  6 
is  the  limit  of  j/  as  x  approaches  1.  Moreover,  it  appears  that  this  limit  is  inde- 
pendent of  the  .succe.ssion  of  values  which  x  assumes  in  approaching  1. 


SLOPE  OF  A  CURVE 


99 


Ex.  2.   Consider  y  = 


as  X  approaches  zero. 


1-vT-x 

Give  X  in  succession  the  values  .1,  .01,  .001,  .0001,  •  •  •.    Then  y  takes  the 
values  1.9487,  1.9950,  1.9995,  1.9999,  •••,  suggesting  the  limit  2. 

In  fact,  by  multiplying  both  terms  of 


VT 


by  1  +  Vl  —  X  we  find 


y  =  1  +  vT^  X  for  all  values  of  x  except  zero. 

Hence  it  appears  that  y  approaches  2  as  x  approaches  0. 

We  shall  use  the  symbol  =  to  mean  "  approaches  as  a  limit." 

Then  the  expressions 

Lim  x  =  a 

and  X  =  a 

have  the  same  significance. 


The  expression 


\Am.f{x)  =  A 


is  read  "  the  limit  oi  f{x),  as  x  approaches  a,  is  A." 

54.  Slope  of  a  curve.  By  means  of  the  conception  of  a  limit 
we  may  extend  the  definition  of  "  slope,"  given  in  §  27  for  a 
straight  line,  so  that  it  may  be 
applied  to  any  curve.  For  let  i^ 
and  ^  be  any  two  points  upon  a 
curve  (fig.  49).  If  ij  and  P^  are 
connected  by  a  straight  line,  the 

slope  of  this  line  is  — ^  •   If  P„ 

and  i^  are  close  enough  together, 

the  straight  line  PyP^  will  differ 

only  a  little  from  the  arc  of  the 

curve,  and  its  slope  may  be  taken 

as  approximately  the  slope  of  the  curve  at  the  point  P^    Now  this 

approximation  is  closer,  the  nearer  the  point  P,  is  to  Py    Hence  we 

are  led  naturally  to  the  following  definition : 

The  slope  of  a  curve  at  a  point  P^ix^,  y^)  is  the  limit  approached 


Fig.  49 


by  the  fraction 


«//(>  ^"~     W-i 


where  x„  and  y.,  are  the  coordinates  of  a 


second  point  P^  on  the  curve,  and  where  the  limit  is  taken  as  P^ 
moves  toward  P^  along  the  curve. 


100 


THE  DERIVATIVE  OF  A  POLYNOMIAL 


Ex.  1.  Consider  the  curve  y  =  x^  and  the  point  (5,  25)  upon  it,  and  let 
Xi  =  5,  yi  =  25. 

We  take  in  succession  various  values  for  x^  and  y^  corresponding  to  points 
on  the  curve  which  are  nearer  and  nearer  to  (xi,  j/i),  and  arrange  our  results  in 
a  table  as  follows  : 


X2 

2/2 

Xo,  -  Xi 

2/2  -  2/1 

2/2-2/1 

Xg-Xi 

6 

36 

1 

11 

11 

5.1 

26.01 

.1 

1.01 

10.1 

6.01 

25.1001- 

.01 

.1001 

10.01 

5.001 

25.010001 

.001 

.010001 

10.001 

The  arithmetical  work  suggests  the  limit  10.    To  verify  this,  place  X2  =  5  +  /;. 

Vi  —  ^1 
Then  ya  =  25  +  10  A  +  A^     Consequently =  10  +  A,  and  as  Xa  approaches 

Xo  —  Xi 

Vn  —  Vi 
Xi,  h  approaches  0  and approaches  10.    Hence  the  slope  of  the  curve 

X2  —  Xi 

y  =  X*  at  the  point  (5,  25)  is  10. 

Ex.  2.    Find  the  slope  of  the  curve  y  =  -  at  the  point  (3,  ^). 


We  have  here 
We  place 


xi  =  3, 


2/1 


Z2  =  3  +  A,   2/2  = 


Then    X2  —  Xi  =  A,  j/2  —  2/1  = 


-h 
9  +  3A 


,  and 


3  +  A 


X2  —  Xi 


9  +  3A 


As  P2  approaches  Pi  along  the  curve,  h  approaches  0,  and  the  limit  of 

^2  —  2/1  ■        1 

— - —  IS  —  -  ;  hence  the  slope  of  the  curve  at  the  point  (3,  ^)  is  —  J. 

In  a  similar  manner  we  may  find  the  slope  of  any  curve  the 
equation  of  which  is  not  too  complicated ;  but  when  the  equation 
is  complicated  there  is  need  of  a  more  powerful  method  for  find- 

ing  the  limit  of  ^ _^-  This  method  is  furnished  by  the  opera- 
tion known  as  differentiation,  the  first  principles  of  which  are 
explained  in  the  following  articles. 

55.  Increment.  When  a  variable  changes  its  value  the  quan- 
tity which  is  added  to  its  first  value  to  obtain  its  last  value 
is  called  its  increment.    Thus  if  x  changes  from   5  to  ^\,  its 


CONTINUITY 


101 


increment  is  ^.  If  it  changes  from  5  to  4|,  the  increment 
is  —  ^.  So,  in  general,  if  x  changes  from  x^  to  x^,  the  increment 
«„— a.%.     It   is   customary    to   denote    an    increment    by   the 


I 


is 

symbol  A  (Greek  delta),  so  that 

Aa  =  x^—  x^,     and     x^  =  x^  +  Ax. 

If  2/  is  a  function  of  x,  any  mcremeut  added  to  x  will  cause 
a  corresponding  increment  of  y.  Thus,  let  y  =f(x),  and  let  x 
change  from  x^  to  x^.    Then  y  changes  from  y^^  to  y^,  where 

2/i  =/K)     and     y„  =f{x.;). 
Hence  Ay  =f(x^)  -f(x^). 

But,  as  shown  above,      x^  =  x^  +  Ax, 
so  that  Ay  =  f{x^  +  Aa:;)  —  f{x^. 

56.  Continuity.  ^  function  y  is  called  a  continuous  function 
of  a  variable  x  when  the  increment  of  y  approaches  zero  as  the 
increment  of  x  approaches  zero. 

It  is  clear  that  a  continuous  function  cannot  change  its  value 
by  a  sudden  jump,  since  we  can  make  the  change  in  the  function 
as  small  as  we  please  by  taking  the  increment  of  x  sufficiently 
small.  As  a  consequence  of 
this,  if  a  continuous  function 
has  a  value  A  when  x  =  a, 
and  a  value  B  when  x  =  'b,  it 
will  assume  any  value  C,  lying 
between  A  and  B,  for  at  least 
one  value  of  x  between  a 
and  h  (fig.  50). 

In  particular,  if  f{a)  is  posi- 
tive and  f{h)  is  negative,  f{x)  =  0  for  at  least  one  value  of  x 
between  a  and  5. 

An  algebraic  polynomial  is  a  continuous  function,  but  we  shall 
omit  the  proof.  The  postage  function  (§20)  is  an  example  of  a 
function  which  is  discontinuous  at  certain  points.  Other  examples 
are  found  in  §§  149,  154. 


x=a 


Fig.  .50 


102  THE  DERIVATIVE  OF  A  POLYNOMIAL 

When  Ax  and  Ay  approach  zero  together  it  usually  happens 

that  —  approaches  a  limit.    In  this  case  y  is  said  to  have  a 

Ax 
derivative,  defined  in  the  next  article. 

57.  Derivative.  When  y  is  a  continuous  function  ofx,the  deriva- 
tive of  y  with  respect  to  x  is  the  limit  of  the  ratio  of  the  increment 
of  y  to  the  increment  of  x,  as  the  increment  of  x  approaches  zero. 

dy 
The  derivative  is  expressed  by  the  symbol  -~  \  or,  if  y  is  expressed 

by  f{x),  the  derivative  may  be  expressed  by  f'{x). 
Th.ViS,iiy=f{x), 

^  =f'(x)  =  Lim  ^  =  Lim  /(^  +  ^/)-/(^) . 
dx  Ax=oA£c      Ax=o  Ax 

The  process  of  finding  the  derivative  is  called  differentiation, 
and  in  carrying  out  the  process  we  are  said  to  differentiate  y  with 
respect  to  x. 

The  process  of  differentiation  involves,  according  to  the  defini- 
tion, the  following  four  steps  : 

1.  The  assumption  of  an  increment  of  x. 

2.  The  computation  of  the  corresponding  increment  of  y. 

3.  The  division  of  the  increment  of  y  by  the  increment  of  x. 

4.  The  determination  of  the  hmit  approached  by  this  quotient, 
as  the  increment  of  x  approaches  zero. 

Ex.  1.    Find  the  derivative  oi  y  =  z^. 

(1)  Assume  Ax  =  h. 

(2)  Compute  Ay  =  {z  +  h)^  -x^  =  Sx^h  +  Sxh^-  +  h^ 

(3)  Find  :^  =  3  x2  +  3  xA  +  K\ 

Ax 

(4)  The  limit  is  evidently  3x2.    Hence  —  =  3x2. 

dz 

Ex.  2.    Find  the  derivative  of  -  • 

X 

(1)  Place  y  =  -  and  assume  Ax  =  h. 

X 

(2)  Compute  Ay         ^  ^  ^ 


X  +  h      X         x2  +  xA 

(8)Find^  = I 

Ax         x2  +  xA 

(4)  The  limit  is  clearly ,  and  therefore  ^  =  -  i. 

x2  dx         x2 


I 


FOKMULAS  OF  DIFFERENTIATION  103 

It  appears  that  the  operations  of  finding  the  derivative  of  f{x) 
are  exactly  those  which  are  used  in  finding  the  slope  of  the  curve 
y  =zf{x).  Hence  the  derivative  is  a  function  wliich  gives  the  slope 
of  the  curve  at  each  point  of  it. 

58.  Formulas  of  differentiation.  The  obtaining  of  a  derivative 
by  carrying  out  the  operations  of  the  last  article  is  too  tedious 
for  practical  use.  It  is  more  convenient  to  use  the  definition  to 
obtain  general  formulas  which  may  be  used  for  certain  classes  of 
functions.  In  this  article  we  shall  derive  all  formulas  necessary 
to  differentiate  a  polynomial. 

1.  — =  naaf'^,  where  n  is  a  positive  integer  and  a  any 

dx 

constant. 

Let  y  =  ax^. 

(1)  Assume       Ax  =  h. 

(2)  Then  Ay  =  a(x+hy— aaf 


=  afnaf-'h  +  ""^"j      ^K''-'7i'+  •  •  ■+h\ 


(3)  ^  =  a(naf-'+'^^''r'^K-'h  +  •  •  •  +  h"-'). 

^  Ax        \  [2  / 


(4)  Taking  the  limit,  we  have  -^  =  naaf  \ 


2.  — — -  =  a,  where  a  is  a  constant. 

dx 

This  is  a  special  case  of  the  preceding  formula,  n  being  here 
equal  to  1.    The  student  may  prove  it  directly. 

dc 

3.  —  =  0,  where  c  is  a  constant. 
dx 

SiDce  c  is  a  constant,  Ac  is   always  0,  no  matter  what  the 

value  of  X.    Hence  —  =0,  and  consequently  the  limit  -j-  =  ^^ 

Ax  <^^ 


104  THE  DERIVATIVE  OF  A  POLYNOMIAL 

4,   The  derivative  of  a  polynomial  is  found  hy   adding   the 
derivatives  of  the  terms  in  order. 

Let  y  =  a^af'-\-a^af-'+--'  +  a^_,x  +  a„. 

(!)  Assume  Ax=h. 

(2)  Then 

Ay  =  a,{x  +  A)"+  a^{x  +  /i)"-'+  •  •  •  +  a„_,{x  +  h)-{-a„ 
-  [a^^af+  a^3f-^+  •  •  •  +  a„_iX  +  aj 

=  h  [naf^af-^  +  {n  —  1)  a^af  "^-j h  a„_^] 

+  ^[n(n-l)a,s(f-'  +  {n-l){n-2)a,af-'-h--'  +  a,^_,] 
H +  h^'a^. 

(3)  ^  =  «aoic"-i  +  (?i  - 1)  a,af -2+  •  •  ■  +  a„_, 
Aa; 


+!()+. ..+A 


n-l. 


"0* 


2 
(4)  Taking  the  limit,  we  have 

-^  =  waoaf-i  +  (w  — l)a,af-^H h  «„_!• 

Ex.    Find  the  derivative  of 

/(x)  =  6x5  -  3a;*  +  5x3 -  7 a;2  +  8x  -  2. 

Applying  formulas  1,  2,  or  3  to  each  term  in  order,  we  have 
/'(x)  =  30x*  -  12x3  +  15x2  -  14x  +  8. 

59.  Tangent  line.  A  tangent  to  a  curve  is  the  straight  line 
approached  as  a  limit  hy  a  secant  line  as  two  points  of  intersection 
of  the  secant  and  the  curve  are  made  to  approach  coincidence. 

It  is  immaterial  in  what  manner  the  two  points  of  intersection 
are  made  to  approach  coincidence.  In  §  37  this  was  done  by 
considering  the  curve  as  moved  in  the  plane.  In  §  88  the  secant 
is  considered  as  moving  parallel  to  itself  until  it  becomes  a 
tangent.  In  this  article  we  are  especially  interested  in  determin- 
ing a  tangent  at  a  known  point  of  the  curve.    Let  us  call  this 


TANGENT  LINE  105 

point  JJ  and  a  second  point  on  the  curve  P,.    Then  if  a  secant  is 

drawn  through  ij  and  i^  of  a  curve  (fig.  51),  and  the  point  F^  is 

made  to  move  along  the  curve  toward  i^,  which 

is  kept  fixed  in  position,  the  secant  will  turn  on 

^  as  a  pivot,  and  wUl  approach  as  a  limit  the 

tangent  F^T.    The  point  F^  is  called  the  point  of 

contact  of  the  tangent. 

From  the  definition  it  follows  that  the  slope 
of  the  tangent  is  the  same  as  the  slope  of  the 
curve  at  the  point  of  contact;  for  the  slope  of  the  tangent  is 
evidently  the  limit  of  the  slope  of  the  secant,  and  this  limit 
is  the  slope  of  the  curve,  by  §  54. 

The  equation  of  the  tangent  is  readily  written  by  means   of 
§  29,  when  the  point  of  contact  is  known.    For,  let  (x^,  y^  be  the 


point  of  contact,  and  let  [-t-]  denote  the  value  of  -7-  when  x  =  x^ 

\"'Vi  ^^  /dy\ 

and  y  =  y^.    Then  {x^,  y^  is  a  point  on  the  tangent  and  ( -p  j  is 

its  slope.    Therefore  its  equation  is  ^     '^ 

The  equation  of  the  tangent  may  also  be  written  in  terms  of 
the  abscissa  of  the  j)oint  of  contact.  Let  a  be  the  abscissa  of  the 
point  of  contact  of  a  tangent  to  a  curve  y  =f{x),  and  let  f{x) 
represent  as  usual  the  derivative  of  f{x).  Then  the  ordinate  of 
the  point  of  contact  is  f{a)  and  the  slope  of  the  tangent  is  f  (a), 
in  accordance  with  §  22.    Hence  the  equation  of  the  tangent  is 

y--f(a)  =  (x-a)fia).  (2) 

Ex.  1.  Find  the  equation  of  the  tangent  to  the  curve  ^^  =  x^  at  the  point 
(Xi,  Vi)  on  it. 

Using  formula  (1),  we  have 

y  -yi  =  Sxl{x-Xi). 

But  since  (xi,  j/i)  is  on  tlie  curve,  we  have  7/1  —  xf.    Tlierefore  the  equation 

can  be  written 

J/  =  3  xf  X  —  2  Xi^ 


106 


THE  DERIVATIVE  OF  A  POLYNOMIAL 


Fig.  52 


Ex.  2.    Find  the  equation  of  the  tangent  to 
y  =  x^  +  Sx 
at  the  point  the  abscissa  of  which  is  2. 
We  will  use  equation  (2).    Then 
/(x)  =  z2  +  3x, 
/(x)  =  2x  +  3. 
/(2)  =  10,  /(2)  =  7. 

Therefore  the  equation  is 

y  — 10  =  7(x  — 2),     or    y  =  7x  — 4. 


If  PT  (fig.  52)  is  a  tangent  line  and  cf>  the  angle  it  m^kes  with 

dy 
OX,'ita  slope  equals  tan  <f),  Ijy  §  28.   Hence  tan  ^  =  —  • 

60.  Sign  of  the  derivative.  A  function  of  x  is  called  an 
increasing  function  when  an  increase  in  x  causes  an  increase  in 
the  function.  A  function  of  x  is 
called  a  decreasing  function  when 
an  increase  in  x  causes  a  decrease 
in  the  function.  The  graph  of  a 
function  runs  up  toward  the  right 
hand  when  the  function  is  increas- 
ing, and  runs  down  toward  the 
right  hand  when  the  function  is 
decreasmg.  Thus  x'—x—  6  (fig.  53) 
is  decreasing  when  «  <  |-,  and  in- 
creasing when  x>'^. 

The  sign  of  the  derivative  enables 
us  to  determine  whether  a  func- 
tion is  increasing  or  decreasing 
in  accordance  with  the  following 
theorem : 

When  the  derivative  of  a  func- 
tion is  positive  the  function  is  in- 
creasing ;  when  the  derivative  is 
negative  the  function  is  decreasing. 

To   prove   this,    consider   y  =/(«),  and    let    us   suppose    that 
dy  .         .  dy  Av  Av 

-r-  is  positive.    Then,  since  ~  is  the  limit  of  -.^ ,  it  follows  that  -r^ 
C'X  cix  Ax  Ax 


SIGN  OF  THE  DERIVATIVE 


107 


is  positive  for  sufficiently  small  values  of  Ax;  that  is,  if  Ax  is 
assumed  positive,  Ay  is  also  positive,  and  the  function  is  increas- 

ing.    Similarly,  if  -r-  is  negative.  Ay  and  Ax  have  opposite  signs 

for  sufficiently  small  values  of  Ax,  and  the  function  is  decreasing 
by  definition. 

dy       ' 
Ex.  1.    If  y  =:  a;2  _  a;  _  6,  —  =  2«  —  1,  which  is  negative  when  x<^  and 

positive  when  x >  ^.    Hence  the  function  is  decreasing  when  x<^  and  increas- 
ing when  x>^,  as  is  sliown  in  fig.  53. 

Ex.2.    If  y  =  |(x3-3x2-9x+27),  Y 

dx      ^  * 


=  |(x  +  l)(x-3). 


Now  —  is  positive  wlien  x  <  —  1, 

negative  when  —  1  <  x  <  3,  and  positive 
when  X  >  3.  Hence  tlie  function  is 
increasing  wlien  x  <  —  1,  decreasing 
when  X  is  between  —  1  and  3,  and 
increasing  when  x  >  3  (fig.  54). 

It  remains    to    examine   the 


cases  in  which  -^  =  0. 
ax 


Eefer- 


FiG.  54 


ring  to  the  two  examples  just 

given,  we  see  that  in  each  the 

values    of   x   which    make   the 

derivative  zero  separate  those  for  which  the  function  is  increasing 

from  those  for  which  the  function  is  decreasing.    The  points  on 

the  graph  which  correspond  to  these  zero  values  of  the  derivative 

can  be  described  as  turning  points. 

Likewise,  whenever  f'{x)  is  a  continuous  function  of  x,  the 
values  of  x  for  which  the  derivative  is  positive  are  separated  from 
those  for  which  it  is  negative  by  values  of  x  for  which  it  is  zero 
(§  56  ).  Now  in  most  cases  which  occur  in  elementary  work 
f'{x)  is  a  continuous  function.    Hence  we  may  say. 

The  values  of  x  for  which  a  function  changes  from  an  increas- 
ing to  a  decreasing  function  are,  in  general,  values  of  x  which 
make  the  derivative  equal  to  zero. 


108  THE  DERIVATIVE  OF  A  POLYNOMIAL 

The  converse  proposition  is,  however,  not  always  true.  A 
value  of  X  for  which  the  derivative  is  zero  is  not  necessarily  a 
value  of  X  for  which  the  function  changes  from  increasing  to 
decreasing  or  from  decreasing  to  increasing.    For  consider 

\{x^-^^+21x-\% 

I         Its  derivative  is  ar^— 6  a?  4-9  =  (a?— 3)^ 

/  which  is  always  positive.    The  func- 

/  tion  is  therefore  always  increasing. 

^  Wlien  a;  =  3  the  derivative  is  zero 

and  the  corresponding  shape  of  tlie 

graph  is  shown  in  fig.  55. 

61.  Maxima  and  minima.     The 

X  turning  points   of   the  graph  of  a 

function   correspond   to   the   maxi- 
mum and  the  minimum  values  of 
the  function.    These  terms  are  more 
precisely  defined  as  follows : 
/(a)  %8  a  maximum  value  of  the  function  f  [x)  when  f  [a  ±h)  <i  f(a) 
for  all  values  of  h  sujficiently  small,  i.e.  for  all  values  of  h  nu- 
merically less  than  some  finite  quantity. 

f{a)  is  a  minimum  value  of  the  function  f  {x)  when  f  {a  ±  h)  >f{a) 
for  all  values  of  h  sufficiently  small. 

In  passing  through  a  maximum  value  the  function  changes 
from  an  increasing  to  a  decreasing  function,  and  in  passing 
through  a  minimum  value  the 'function  changes  from  a  decreas- 
ing to  an  increasing  function.  From  the  work  of  the  previous 
article  we  may  accordingly  frame  the  following  rule  for  finding 
the  maxima  and  the  minima  values  of  a  function : 

Find  the  derivative  of  the  function,  place  it  equal  to  zero,  and 
solve  the  resulting  equation.  Take  each  root  thus  found  and  see 
if  the  derivative  has  opposite  signs  as  x  is  taken  first  a  little 
smaller  and  then  a  little  larger  than  the  root.  If  the  sign  of  the 
derivative  changes  from  plus  to  mimis,  the  root  substituted  in  the 
fwnetion  gives  a  maximum  value  of  the  function.  If  the  sign  of 
the  derivative  changes  from  minus  to  plus,  the  root  suhstituted  in 
the  function  gives  a  minimum  value  of  the  function. 


MAXIMA  AND  MINIMA 


109 


This  rule  is  most  readily  applied  when  the  derivative  can  be 
factored.  The  change  of  sign  is  then  determined  as  in  §  46. 
In  §  62  will  be  given  a  method  of  distinguishing  between  a  maxi- 
mum and  a  minimum,  which  may  be  used  when  the  factoring  of 
the  derivative  is  not  convenient.  In  practical  problems  the  ques- 
tion as  to  whether  a  value  of  x  for  which  the  derivative  is  zero 
corresponds  to  a  maximum  or  a  minimum  can  often  be  deter- 
mined by  the  nature  of  the  problem. 

Ex.  1.    Find  the  maximum  and  the  minimum  values  of 

/(x)  =  xs  _  5x* -f- 5x3  +  10a;2  -  20a;  +  5. 
We  find  /(x)  =  5 x*  -  20x3  +  15x2  -|-  20x  -  20 

=  5(x2-l)(x2-4x  +  4) 
=  5(x  +  l)(x-l)(x-2)2. 

Tlie  roots  of  /'  (x)  =  0  are  —  1,  1,  and  2.  As  x  passes  through  —  1,  /'(x) 
changes  from  +  to  — .  Hence  x  =  —  1  gives  /(x)  a  maximum  value,  namely  24. 
As  X  passes  through  +  1,  /"(x)  changes  from  —  to  +.  Hence  x  =  +  1  gives  /(x) 
a  minimum  value,  namely  —  4.  As  x  passes  through  2,  /'  (x)  does  not  change 
sign.    Hence  x  =  2  gives  /(x)  neither  a  maximum  nor  a  minimum  value. 

Ex.  2.  A  rectangular  box  is  to  -be  formed  by  cutting  a  square  from  each 
corner  of  a  rectangular  piece  of  cardboard  and  bending  the  resulting  figure. 
The  dimensions  of  the  piece  of  cardboard  being  20  by  30  inches,  required  the 
largest  box  which  can  be  found. 

Let  X  be  the  side  of  the  square  cut  out.  Then  if  the  cardboard  is  bent  along 
the  dotted  lines  of  fig.  56,  the  dimensions  of  the  box  are  30  —  2  x,  20  —  2  x,  z. 
Let  y  be  the  volume  of  the  box.    Then 

y  =  X  (20  -  2  x)  (30  -  2  x) 
=  600x- 100x2 +  4x3. 
dy 


dx 


=  600  -  200  x  + 12x2. 


Equating  this  to  zero,  we  have 
3x2  _60x  + 150  =  0, 

25  ±  5  V? 


Hence 


dy 
dx 


3.9  or  12.7. 


12(x-3.9)(x-12.7). 


a- 

X 

[                    30-2X 

Ti 

?! 

1 

Fig.  56 


dx 


changes  from  +  to  —  as  x  passes  through  3.9.    Hence  x  =  3.9  gives  the 

maximum  value  1056+  for  the  capacity  of  the  box.  x  =  12.7  gives  a  mini- 
mum value  of  ?/,  but  this  has  no  meaning  in  the  problem  for  which  x  must 
lie  between  0  and  10. 


110  THE  DERIVATIVE  OF  A  POLYNOMIAL 

Ex.  3.    The  deflection  of  a  girder  resting  on  three  equally  distant  supports 
and  loaded  uniformly  is  given  by  the  equation 

V  =  C (-  l^  +  Slx^  -  2x*), 

where  C  is  a  constant,  I  the  distance  between  the  supports,  and  x  the  distance 
from  the  end  support.    Required  the  point  of  maximum  deflection. 

dz 

Equating  this  to  zero,  we  have 

8x3-9^2  +  18  =  0. 

It  is  clear  that  in  the  practical  problem  x<l.    We  find  by  trial  that  a  root 
lies  between  x  =  Al  and  x  =  .51.    We  will  place 

y  -Sx^  -dlx^  +  P, 

and  apply  the  method  of  §  62.    The  straight  line  connecting  (.41,  .072^3)  and 

(.5L  -.25i3)is 

y  -  .072  Z3  =:  _  3.22  i2  (X  -  .4  0 

and  this  cuts  the  axis  of  x  when 


-(■^-Si)'=- 


This  is  approximately  the  root  of  the  equation.  As  a  check  we  note  that  when 
X  =  .42  i,  y  =  .006104 1^ ;  and  when  x  =  .43  i,  y  =  -  .028044 1\  Hence  the  root 
lies  between  .42 1  and  .43  J. 

If  more  accuracy  is  required,  the  straight  line  connecting  (.42  Z,  .006104^3) 
and  {A31,  —  .028044  i^)  may  be  found.    Its  intercept  on  OX  is 

x=.4215Z. 

As  shown  in  §  63,  Ex.  2,  this  is  correct  to  four  decimal  places. 

62.  The  second  derivative.    Since  -^  is  in  general  a  function  of 

ax 

X,  it  may  be  differentiated  with  respect  to  x.  The  result  is  called 
the  second  derivative  of  y  with  respect  to  x,  and  is  indicated  by 

the  symbol  -r-  (-r-)'  which  is  commonly  abbreviated  into  — ^  • 
''  dx\dx/  ^  dod^ 

When  a  function  is  denoted  by  f{x)  and  its  derivative  by  f'{x), 

its  second  derivative  is  denoted  by  f"{x) ;  thus,  if 

^= /'(»)=  3a^- 6  a; +6, 
ax 

0=/"(.)=6.-6. 


THE  SECOND  DEKIVATIVE 


111 


Again,  by  differentiating  — ^  or  f"{x),  we  may  obtain  an  expres- 
sion  called  the  third  derivative,  denoted  by  — ^  or  f"'{x).  By  dif- 
ferentiating this  we  obtain  the  fourth  derivative,  and  so  on.    To 

distinguish  -^  from  these  higher  derivatives  it  is  sometimes  called 
ax 

the  first  derivative.     ^ 

The  significance  of  — 4  for  the  graph  is  obtained  from  the  fact 

dy  d^y 

that  -^  is  equal  to  the  slope ;  hence  — ^  is  the  derivative  of  the 

CtX  72  clx 

slope.    Therefore,  by  §  60,  if  — ^  is  positive,  the  slope  is  increas- 

d^y 
ing ;  if  — ^  is  negative,  the  slope  is  decreasing.    We  may  have, 

accordingly,  the  following  four  cases : 


1.  ^  is  +, 
dx 


dy? 


IS  +. 


The  graph  runs  up  toward  the  right  with 
increasing  slope  (fig.  57). 


2.  "f.  is  +, 
dx 


d'y, 


da? 


IS  — . 


The  graph  runs  up  toward  the  right  with 
decreasing  slope  (fig.  58). 


3.  "f  is  -, 

dx 


da? 


IS  +. 


The  graph  runs  down  toward  the  right. 
The  slope  which  is  negative  is  increasing 
algebraically  and  hence  is  decreasing 
numerically  (fig.  59). 


4.^  is 
dx 


da? 


IS 


Tlie  graph  runs  down  toward  the  right 
and  the  slope  is  decreasing  algebraically 
(fig.  60). 


Fig.  60 


112 


THE  DERIVATIVE  OF  A  POLYNOMIAL 


The  consideration  of  these  types  leads  to  the  following  con- 
clusion '■  If  ^^  positive,  the  graph  is  concave  upward ;  if  — ^ 


d3? 


da? 


is  negative,  the  graph  is  concave  dovmward. 


From  this  we  may  deduce  the  following  rule  to  distinguish 
maxima  and  minima  in  that  we  take  accoimt  of  the  fact  that 
the  graph  is  concave  upward  when  y  is  a  minimum  and  concave 

rj.  dy  .  -,  d?y  . 

downward  when  y  is  a  maxunum.    Ij  -f-  u  zero  and  — ^  %s  posv- 

tive,  y  has  a  minimum  value  ;  ij  —  is  zero  and  -77^  is  negative,  y 
has  a  m,aximum  value.  1 

This  rule  cannot  be  applied  to  the  case  in  wliich  ~r~^  ^^^ 

—  =  0,  and  hence  it  is  not  so  complete  as  the  rule  in  §  61,  but  it 
da? 

is  sometimes  more  convenient  in  application,  and  especially  when 
the  first  derivative  cannot  be  factored. 

When  the  curve  changes  from  concavity  in  one  direction  to  con- 

cavity  in  the  other,  — ^  =  0.    Tlie  corresponding  point  is  called  a 

point  of  inflection.    Hence  to  find  the  points   of  inflection  we 

must  solve  the  equation  — -^  =  0,  and  see  if  the  second  derivative 

changes  sign  as  x  passes  through 
each  root. 


Ex.  1.    y^^^ix^-Qx"^), 


dx 


x  =  -x{x-4), 


dy 


Fig.  61 


d'^y 


d^i/      1  1 

^^  =  -x-l  =  -(x-2). 

dx^      2  2^  ' 

The  curve  (fig.  61)  is  concave  down- 
ward when  X  <  2,  is  concave  upward 
when  a;  >  2,  and  has  a  point  of  inflec- 
tion when  x  =  2.    When  x  =  0,  —  =  0 

fPy  dx 

and  -—  <  0 ;  the  corresponding  value 

of  y  is  therefore  a  maximum.    When 


«  =  4,  -—  =  0  and  "  ^  >  0 ;  the  corresponding  value  of  y  is  therefore  a  minimum. 


EXAMPLES 


113 


3  x2  +  o, 


Ex.  2.    y  =  x^  +  ax  =  x(x^  +  a), 
dy 
dx 

dx2 

The  curve  is  concave  downward  when  x  <  0,  is  con- 
cave upward  wlien  x  >  0,  and  has  a  point  of  inflection 
when  X  =  0.    In  addition  we  distinguish  two  cases : 


(1)  a  positive. 


dy 
dx 


is  always  positive,  and  the  curve 


cuts  OX  only  at  the  origin  (fig.  62). 

(2)  a  negative.    The  curve  has  a  maximum  ordinate 


when  X  = 


I 


and  has  a  mini- 


mum ordinate  when 


H- 


y  =  + 


2a 


Fig.  62 


0,  or  -H  V  —  a 


It  cuts  OX  when  x 
(fig.  63). 

Ex.  3.    y  =  x^  +  ax  +  b. 

The  graph  of  this  function  may  be  obtained 
by  moving  the  graph  of  Ex.  2  through  the  dis- 
tance b  up  or  down,  according  to  the  sign  of  b. 
Our  interest  is  especially  with  the  intei'cepts 
on  OX.  The  curve  obtained  from  (1)  of  Ex.  2 
cuts  the  axis  of  x  in  one  and  only  one  point. 
The  curve  obtained  from  (2)  of  Ex.  2  will 
intersect  OX  in  three  points,  will  intersect 
OX  in  one  point  and  be  tangent  in  another, 
or  will  intei'sect  OX  in  one  point  only,  accord- 
ing as  the  numerical  value  of  b  is  less  than, 
equal  to,  or  greater  than  the  distance  of  the 
tui'ning  point  of  the  curve  from  OX ;  that  is, 
according  as 

b^  = 


2a     r^a\ 
Y\      3/ 


This  condition  reduces  to 

62 


Fig.  03 


4       27 


0. 


114 


THE  DERIVATIVE  OF  A  POLYNOMIAL 


{,2  qS 

It  is  to  be  noticed  that  when  a  >  0,  -  +  —  >  0.    Hence  we  may  cover  all 

4        27 
cases  by  the  statement : 

The  eauation  x^  +  ax  +  b  =  0  has  three  unequal  reed  roots,  two  equal  real  roots 

52      a'  < 
and  one  other  real  root,  or  one  real  and  two  complex  roots,  according  aa  -  +  —  =  0. 

63.  Newton's  method  of  solving  numerical  equations.  The 
results  of  this  chapter  may  be  applied  to  finding  approximately 
the  irrational  roots  of  a  numerical  equation.  We  first  find,  by  the 
method  of  §  47,  two  numbers  x^  and  x^,  between  which  a  root 
of /(a:)=0  is  known  to  lie.  It  is  necessary  to  take  care  that 
neither /'(a;)  norf"{x)  is  zero  for  any  value  of  x  between  x^^  and  x^. 
Then  f{x)  is  always  increasing  or  decreasing  between  x^  and  x^ 
and  hence  only  one  root  of  f{x)  =  0  lies  between  x^  and  x^.    Also 


N 


Xi  D/  /C 


U 


(1) 


D       X2 


(S) 


N 


Fig.  64 


the  curve  y= /(a?)  is  always  concave  upward  or  concave  down- 
ward between  aj^  and  x^.  Hence  the  curve  has  one  of  the  four 
shapes  of  fig.  64. 

It  appears  that  in  each  case  a  tangent  at  one  of  the  points 
M  ox  N  will  intersect  the  axis  of  ic  in  a  point  C  which  lies 
between  x^  and  x^.  In  practice  it  is  most  convenient  to  sketch 
the  curve  with  attention  to  the  signs  of  the  first  and  the  second 
derivative,  and  to  find  the  tangent  at  that  end  at  which  it  lies 
between  the  curve  and  the  ordinate  of  the  point  of  contact. 
The  intersection  of  the  tangent  with  OX  is  then  nearer  to  the 


J 


I 


NEWTON'S  METHOD  115 

intersection  of  the  curve,  i.e.  to  the  required  root  of  the  equation, 
than  is  the  abscissa  of  the  point  of  contact.  For  example,  in 
fig.  64,  (1)  and  (4),  the  equation  of  the  tangent  is 

fix ) 
and  its  point  of  intersection  with  OX  is  x  —  •^ ,    ^   •    Hence  the 

root  which  was  at  first  known  to  lie  between  x^  and  x^  is  now 

fix ) 
known  to  lie  between  x.  and  x„  —  ^;  ^'  • 

It  is  well  in  practice  to  combine  this  method  with  the  method 
of  §  52.  For,  if  we  draw  the  secant  MN,  it  will  intersect  the 
axis  of  ic  in  a  point.  D,  and  the  root  of  the  equation  lies  between 
C  and  D.  But  C  and  D  are  closer  together  than  are  x^  and  x^,  so 
that  we  have  narrowed  down  the  interval  within  which  the  root  lies. 

Ex.  1.    Find  the  root  of  a;^  —  6  x  —  13  =  0,  which  lies  between  3  and  4. 

Here  /(a;)  =  x^  -  6  x  -  13, 

/'(x)  =  3x2-6, 
/"(x)=6x. 

When  X  =  3,/(x)  =  —  4  ;  and  when  x  =  4,/(x)  =  27  ;  while  between  x  =  3  and 
X  =  4,  f'{x)  and  f"{x)  are  positive.  Hence  the  graph  is  as  in  fig.  64,  (1),  where  M 
is  (3,  -  4)  and  N  is  (4,  27).    The  tangent  at  N  is 

y  -21  =  42(x-4). 
Hence,  for  C,  x  =  4-2j  =  3.36. 

The  equation  of  MN  is        y  -  27  =  31  (x  -  4). 
Hence,  for  D,  x  =  4  -  §^  =  3. 13. 

Therefore  the  root  lies  between  3.13  and  3.36. 

As  this  does  not  fix  the  first  decimal  figure  of  the  root,  it  is  advisable  to  apply 
§  47  again.  We  find  /(3. 1)  =  -  1.809  and  /(3.2)  =  +  .568.  Hence  the  root  lies 
between  3.1  and  3.2.  Accordingly,  the  point  M  is  now  (3.1,  —  1.809),  and  the 
point  N  is  (3.2,  .568).    The  equation  of  the  tangent  at  N  is 

y-  .568  =  24.72(x-3.2), 
and  for  the  new  point  C  x  —  3.17702. 

The  secant  MN  is  y  -  .568  =  23.77  (x  -  3.2) 

and  for  D  x  =  3.176. 

The  root  of  the  equation  therefore  lies  between  3.176  and  3.177.  This  result 
is  close  enough  for  most  practical  purposes,  but  if  the  operations  are  carried 
out  once  more  it  is  found  that  the  root  lies  between  3.1768148  and  3.1768144. 


116  THE  DERIVATIVE  OF  A  POLYNOMIAL 

Ex.2.  In  §61,  Ex.3,  the  root  of  8x^ -Olx"^  +  1^  =  0  was  found  to  lie 
between  A21  and  .43  Z, 

Placing  /(a;)  =  8x8-9ix2  +  Z3, 

we  have  /'(x)  =  24  a;-  -  18  Ix, 

/"(x)  =  48x-18Z, 

so  that/'(x)  is  negative  and  /"(x)  positive,  when  x  is  between  A21  and  AS  I. 
Hence  the  curve  has  the  shape  of  fig.  64,  (3).  The  tangent  at  (.42  Z,  .005104^3) 
meets  OX  where  x  =  .42153 1.  The  chord  connecting  (.42 1,  .005104  P)  and  (.43 1, 
-  ,028044 1^)  meets  OX  where  x  =  .42154  Z.  The  root  is  therefore  determined  to 
four  decimal  places. 

64.  Multiple  roots  of  an  equation. 

If       f{x)  =  aQ3if+a^xr-^+a^x"-^-] \- a„_^os^-{- a„_iX  + a^, 

f'{x)  =  na^af'-'^  +  {n  -  l)a^c(f'-''  +  {n  -  2)a^af-'^-\ 

+  2  a„_^x  +  a„_^, 

f\x)  =  n(n-  l)a^ar-'+{n  -l){n-  2)a^o^-^ 
+  {n-2){n-  3) a^af -*+  •  •  •  +  2  a„_j, 

f"'{x)  =  n{n-l){n-2)  a^-" 

+  {n-l){n,-2){n  —  S)a^af-*-] , 

and  so  on.  Now  let  /(a),  /'(a),  f"{a),  /'"{a),  etc.,  denote  the  result 
of  placing  x  =  a  va.  these  functions,  and  f{a  +  h)  denote  the  result 
of  placing  x  =  a  +  hin  f{x).    One  readily  computes  that 

f{a  +  ^)  =  f{a)  +  hf'(a)  +  ^  f"{a)  +  ^  f"'(a)  + . . .  +  «^».  (1) 

In  (1)  place  h  =  x  —  a  and  it  becomes 

f{x)  =:f(a)  +  {x-  a)f'{a)  +  ^^^V"(«) 

+  ^^^/'»+---  +  «o(^-<.  (2) 

If  now  a  is  a  double  root  oif{x)  =  Q,f{x)  is  divisible  by  {x—af, 
by  §  42,  and  therefore,  by  (2),  f{a)  =  0,  f{a)  =  0.  If  a  is  a  triple 
root  of  f{x)  =  0,  /(.>:)  is  divisible  by  (x—  a)\  and  therefore  /(a)  =  0, 
f'(a)  =  0,  /"  (a)  =  0.  Similar  statements  may  be  made  for  multiple 
roots  of  higher  order. 


MULTIPLE  KOOTS 


117 


Conversely,  if  f{a)  =  0  and  f'{ci)  =  0,  (2)  shows  that  f(x)  is 
certainly  divisible  by  {x  —  aY  and  perhaps  by  a  higher  power  of 
X  —  a.  Therefore  a  is  a  multiple  root  of  f(x)  =  0.  We  have  then 
the  result: 

A  multiple  root  of/{x)  =  Ois  also  a  root  off'{x)=0,  and  conversely. 

Hence  we  may  find  the  multiple  roots  of  f{x)  =  0  by  equating 
to  zero  the  highest  common  factor  of  f{x)  and  f'{x)  and  solving 
the  resulting  equation. 

The  condition  that  an  equation  f{x)  =  0  should  have  multiple 
roots  is  the  vanishing  of  the  discriminant  of  the  equation,  which 
is  the  eliminant  of  the  equations  /{x)  =  0  and  f'{x)  =  0,  and  may 
be  found  by  the  method  of  §  9. 

Ex.  1.    Find  the  discriminant  of  ax^  +  6x  +  c  =  0. 
We  have  to  find  the  condition  that  the  two  equations 

ax2  +  6x  +  c  =  0 
and  2  ax  +  6  =  0 

should  have  a  common  root.    Multiplying  the  last  equation  by  x,  we  have 

2  ax2  +  6x  =  0, 

and  the  determinant  of  the  coefficients  and  the  absolute  terms  of  the  three 

equations  is 

a  b  c 

0  2  a  6=0, 

2a  &  0 

62  _  4  ac  =  0. 

Ex.  2.    Find  the  discriminant  of  x^  +  ox  +  6  =  0. 
We  must  find  the  eliminant  of  this  and 

3x2  + a  =  0. 

Multiplying  the  first  equation  by  x,  and  the  second  by  x  and  x^,  we  have  the 

five  equations 

X*        +  ax2  +  6x         =  0, 

x3  +  dx  +  6  =  0, 

3x*       +  ax2  =0, 

3x8  +ax         =0, 

3x2  +a  =  0, 

1  0  a  6    0 

0  1  0  a    6 

3  0  a  0    0=0, 

0  3  0  a    0 

0  0  3  0a 

4a3  +  2762  =  0.     (See  §  62,  Ex.  3.) 


■and  their  eliminant  is 


118  •       THE  DERIVATIVE  OF  A  POLYNOMIAL 

PROBLEMS 

Find  the  respective  slopes  of  the  following  curves  at  the  points  noted: 
(1)  by  an  approximate  numerical  calculation,  as  in  §  54 ;  (2)  by  placing  x  equal 
to  the  abscissa  of  the  given  point,  plus  A,  and  allowing  h  to  approach  zero : 

1.  y  =  a;3  at  (2,  8). 

2.  y  =  a;2  -  3  X  at  (0,  0). 

3.  y  =  a;3_3x  +  l  at  (1,  -1). 

4.  Find  the  derivative  of  x^  —  a;  by  using  the  definition  but  not  the  formulas. 

5.  Find  the  derivative  of  3  x*  +  2  x  by  using  the  definition  but  not  the 
formulas. 

Find  the  derivative  of  each  of  the  following  expressions  by  the  fonnulas : 

6.  |X6  -|X5  +  X. 

7.  4x8-6x2  +  5x-8. 

8.  6x9-6x8  +  7x«-4x4-2x2  +  3x-9. 

9.  By  expanding  and  differentiating  show  that  the  derivative  of  (3  x  +  2)* 
isl2(3x  +  2)8. 

10.  By  expanding  and  differentiating  show  that  the  derivative  of  (x  +  a)"  is 
n(x  +  a)»-i. 

11.  Find  the  equation  of  the  tangent  to  the  curve  i/  =  x*  +  3  at  the  point  the 
abscissa  of  which  is  —  2. 

12.  Show  that  the  equation  of  the  tangent  to  the  curve  y  =  x^  +  ax  +  6  at 
the  point  (Xi,  i/i)  is  y  =  (3x,''  +  a)x  -  2xf  +  6. 

13.  Show  that  the  equation  of  the  tangent  to  the  curve  y  =  ax-  +  2  6x  +  c  at 
the  point  (xi,  yi)  is  y  =  2(axi  +  b)x  —  axf  +  c. 

14.  Determine  the  point  of  intersection  of  the  tangents  to  the  curve  y  = 
x^  —  6x  +  7  at  the  points  the  abscissas  of  which  are  —  2  and  3  respectively. 

15.  Find  the  angle  between  the  tangents  to  the  curve  y  =  2x2  —  3x  +  lat 
the  points  the  abscissas  of  which  are  —  1  and  2  respectively. 

16.  Find  the  area  of  the  triangle  included  between  the  coordinate  axes  and 
the  tangent  to  the  curve  y  =  x^  at  the  point  (2,  8). 

17.  Find  the  points  on  the  curve  y  =  x*  —  3x  +  7at  which  the  tangents  are 
parallel  to  the  line  y  =  dx  +  S. 

18.  How  many  tangents  has  the  curve  y  =  x8— 4x2  +  x  —  4  which  are 
parallel  to  the  line  y  +  4x  +  7=0?    Find  their  equations. 

19.  Find  the  points  on  the  curve  y  =  x^  +  x^  —  6  at  which  it  makes  an  angle 
of  45°  with  OX. 


PROBLEMS  119 

Find  the  values  of  x  for  which  the  following  expressions  are  respectively 
increasing  and  decreasing: 

20.  x2  +  4a;-7.  '  22.  a;*  +  8 x  -  10. 

21.  x3-2x2  +  8.  23.  X*- 2x2  + 6. 

24.  Find  the  lowest  point  of  the  curve  y  =  3x^  —  8x  +  l, 

25.  Find  the  turning  points  of  the  curve  ?/  =  i  x*  —  2  x2  +  i. 

Find  the  maximum  and  the  minimum  values  of  the  following  expressions  : 

26.  3x8  _  2x2  _  5a;  +  1.  27.  3x5  -  25x3  +  60x  -  50. 

28.  Prove  that  the  largest  rectangle  with  a  given  perimeter  is  a  square. 

29.  A  rectangular  piece  of  cardboard  a  in.  long  and  b  in.  broad  has  a  square 
cut  out  of  each  corner.  Find  the  length  of  a  side  of  this  square  when  the  box 
formed  from  the  remainder  has  its  greatest  volume. 

30.  Find  the  dimensions  of  the  greatest  rectangle  which  can  be  inscribed  in 
a  given  isosceles  triangle  with  base  b  and  altitude  h. 

3 1 .  Find  the  right  circular  cylinder  of  greatest  volume  which  can  be  inscribed 
in  a  sphere  of  radius  a. 

32.  Find  the  right  circular  cylinder  of  greatest  volume  which  can  be  cut  from 
a  given  right  circular  cone. 

•     33,  Find  the  point  of  the  line  3x  +  y  =  Q  such  that  the  sum  of  the  squares 
of  its  distances  from  the  two  points  (5,  1)  and  (7,  3)  may  be  a  minimum. 

34.  Among  all  circular  sectors  with  a  given  perimeter  find  the  one  which 
has  the  greatest  area. 

35.  A  rectangular  box  with  a  square  base  and  open  at  the  top  is  to  be  made 
out  of  a  given  amount  of  material.  If  no  allowance  is  made  for  thickness  of 
material  or  waste  in  construction,  what  are  the  dimensions  of  the  largest  box 
that  can  be  made  ? 

36.  A  length  I  of  wire  is  to  be  cut  into  two  portions,  which  are  to  be  bent 
into  the  forms  of  a  circle  and  a  square  respectively.  Show  that  the  sum  of  the 
areas  of  these  figures  will  be  least  when  the  wire  is  cut  in  the  ratio  ir  :  4. 

37.  A  piece  of  galvanized  iron  b  ft.  long  and  a  ft.  wide  is  to  be  bent  into  a 
U-shaped  water  pipe  b  ft.  long.  If  we  assume  that  the  cross  section  of  the  pipe  is 
exactly  represented  by  a  rectangle  on  top  of  a  semicircle,  what  are  the  dimensions 
of  the  rectangle  and  the  semicircle  that  the  pipe  may  have  the  greatest  capacity  ; 
(1)  when  the  pipe  is  closed  on  top  ?  (2)  when  it  is  open  on  top  ? 

38.  A  stream  flowing  with  the  velocity  a  strikes  an  undershot  water  wheel, 
giving  it  the  velocity  x.  Assuming  that  the  efficiency  of  the  wheel  is  propor- 
tional to  the  velocity  x  of  the  wheel  and  the  loss  of  velocity  a  —  x  of  the 
water,  what  is  the  velocity  of  the  wheel  when  it  has  its  greatest  eflBciency  ? 


120  THE  DERIVATIVE  OF  A  POLYXOMIAL 

39.  A  gardener  has  a  certain  length  of  wire  fencing  w4th  which  to  fence 
three  sides  of  a  rectangular  plot  of  laud,  the  fourth  side  being  made  by  a  wall 
already  constnicted.  Required  the  dimensions  of  the  plot  which  contains  the 
maximum  area. 

40.  For  a  continuous  girder  of  uniform  section,  uniformly  loaded,  and  con- 
sisting of  three  equal  spans,  the  deflection  in  the  middle  span  is  given  by  the 
equation  v  —  C  {l^x  —  6  l^x-  +  lOlx^  —  5x*),  where  C  is  constant,  I  the  length  of 
the  span,  and  x  the  distance  from  a  point  of  support.  Find  the  greatest 
deflection. 

41.  If  p  is  the  density  of  water  and  t  the  temperature  between  0°  and  30°  C, 
p  =  po(l  +  lt  +  mt^  +  vi^),  where  po  is  the  density  when  t  =  0,  and  /  =  .000052939, 
m  =  -  .0000065322,  n  =  .00000001445.  Show  that  the  maximum  density  occure 
when  «  =  4.108°. 

42.  Show  that  the  curve  y  =  ax^  +  bx  +  c  is  concave  upward  or  downward 
according  as  a  is  positive  or  negative. 

43.  Show  that  the  curve  y  =  x^  +  ax  +  b  is  concave  upward  when  x  is  posi- 
tive and  concave  downward  when  x  is  negative. 

Determine  the  values  of  x  for  which  the  foUowhig  curves  are  concave 
upward  or  downward  : 

44.  y  =  x8-3x2-24.  45.  y  =  a;5_5a;  +  6. 
Find  the  points  of  inflection  of  the  following  curves : 

46.  Gy  =  x^-Gx^-\-Gx  +  1.  47.   12y  =  x*  -  6x^  +  12x^  -  2x+ 1. 

48.  y  =  3x^-  lOx*  +  10x8  +  6x  -  8. 

49.  y  =  3x5  -  5x*  4-  20x8  -  60x2  +  20x  -  5. 

50.  Prove  that  the  curve  y  =  ax^  +  bx^  +  ex  +  d  always  has  one  and  only 
one  point  of  inflection. 

Find  the  real  roots  of  the  following  equations  accurate  to  two  decimal 
places : 

51.  x8  -  x2  -  2x  +  1  =0.  54.  x^  -  3x2  -  2x  +  5  =  0. 

52.  x8  +  3x2  +  4x  +  5  =  0.  55.  x*  -  x^  -  x2  +  x  -  1  ==  0. 

53.  x8-2x  -5  =  0. 

Show  that  each  of  the  following  equations  has  equal  roots  and  solve  it : 
56.  x8-x2-8x+  12  =  0.  57.  x*  -  2(l-a)x8  +  (l_  3a)x2  +  a  =  0. 

Find  the  condition  that  each  of  the  following  equations  should  have  equal 
roots  : 

58.  x8  +  3  ax2  +  6  =  0.  60.  x*  +  4  ax  +  6  =  0. 

59.  X*  +  4ax8  +  6  =  0.  61.  aox3  +  3aix2  +  Soox  +  as  =  0. 


CHAPTEE  VI 
CERTAIN  ALGEBRAIC  FUNCTIONS  AND  THEIR  GRAPHS 

65.  Square  roots  of  polynomials.  In  the  previous  chapters  the 
discussion  has  been  restricted  to  the  polynomial.  We  will  next 
study  the  square  root  of  the  polynomial. 

At  first  let  us  assume  that  the  polynomial  can  be  separated 
into  71  linear  real  factors,  as  in  §  42.    We  have,  then, 


y 


=  ±  Va,  (X  -  r,)  {x-r^)---{x-  r„),  (1) 


and  the  graph  of  this  function  can  readily  be  constructed  by  con- 
sidering the  graph  of 

y  =  «o  (^  -  ^i)  {-^  -  ^^2)  •••(•»-  r„),  (2) 

as  given  in  §  46. 

In  the  first  place,  the  graph  of  (1)  will  intersect  the  axis  of  ic  in  the 
same  points  as  the  graph  of  (2),  i.e.  in  the  points  x  =  1\,  x  =  r^,  •  •  •, 
as  for  these  values  of  x  the  product  under  the  radical  sign  is  zero. 

In  the  second  place,  wherever  the  graph  of  (2)  is  below  the 
axis  of  X,  the  expression  under  the  radical  sign  in  (1)  is  negative, 
tlie  value  of  the  radical  is  imaginary,  and  hence  there  is  no  cor- 
responding point  of  the  graph.  If,  however,  the  graph  of  (2)  is 
above  the  axis  of  x,  there  are  two  values  of  y  in  (1),  equal  in 
magnitude  and  opposite  in  sign,  and  correspondingly  there  are  two 
points  of  the  graph  situated  symmetrically  with  respect  to  OX. 
Therefore  OX  is  an  axis  of  symmetry. 

As  the  negative  values  of  the  expression  under  the  radical  sign 
are  separated  from  the  positive  values  by  zero,  it  follows  that  the 
values  of  x  which  make  the  expression  zero,  i.e.  r^,r„,---,  r„,  are  of 
the  utmost  importance  in  plotting  these  graphs.  In  fact,  the  lines 
X  =  1\,  x=.r^,---,x  =  r^  divide  the  plane  into  sections  bounded  by 
straight  lines  parallel  to  0  F,  in  which  there  will  be  no  part  of  the 

121 


122 


CEKTAIN  ALGEBRAIC  FUNCTIONS 


graph  if  the  corresponding  values  of  x  make  the  expression  negative, 
and  in  which  there  will  be  a  part  of  the  graph  if  the  corresponding 
values  of  x  make  the  expression  positive.  Hence  the  first  step  in 
plotting  the  graph  is  the  drawing  of  these  lines  and  the  determi- 
nation of  which  sections  of  the  plane  should  be  considered. 


Ex.  \.    y  =  ±  V(x  +  2)  (X  -  1)  (X  -  5). 

K  X  =  —  2,  1,  or  5,  y  =  0,  and  the  graph  intersects  the  axis  of  x  at  three  points. 

The  lines  x  =  —  2,  x  =  l,  x=5  divide  the  plane  (fig.  65)  into  four  sections. 

If  a;  <  —  2,  all  three  factors  of  the  product  are  negative ;  hence  the  radical 
is  imaginary  and  there  can  be  no  part  of  the  graph  in  the  corresponding  section 
of  the  plane. 


Fig.  65 


If  -  2  <  X  <  1,  the  first  factor  is  positive  and  the  other  two  are  negative  ; 
hence  the  radical  is  real  and  there  is  a  part  of  the  graph  in  the  corresponding 
section  of  the  plane. 

If  1  <  X  <  5,  the  first  two  factors  are  positive  and  the  third  is  negative  ; 
hence  the  radical  is  imaginary  and  there  can  be  no  part  of  the  graph  in  the 
corresponding  section  of  the  plane. 


GRAPHS 


123 


Finally,  if  x  >  5,  all  three  factors  are  positive ;  hence  the  radical  is  real  and 
there  is  a  part  of  the  graph  in  the  corresponding  section  of  the  plane. 

Therefore  the  graph  consists  of  two  separate  parts,  and  is  seen  (fig.  65)  to 
consist  of  a  closed  loop  and  a  branch  of  infinite  length. 


Ex.  2.    y  =  ±  V(x  +  4)  (X  +  2)  (x  -  1)  (x  -  4). 

If  X  =  —  4,  —  2,  1,  or  4,  y  =  0,  and  the  graph  intersects  the  axis  of  x  at 
four  points. 

The  lines  x  =  —  4,  x=  —  2,  x  =  l,  and  x  =  4  divide  the  plane  (fig.  66)  into 
five  sections. 


I 


-X 


Fig.  66 

If  X  <  -  4,  all  four  factors  are  negative  ;  hence  the  radical  is  real  and  tlaere 
is  a  part  of  the  graph  in  the  first  section. 

If  -  4  <  X  <  -  2,  the  first  factor  is  positive  and  the  others  are  negative ; 
hence  the  radical  is  imaginary  and  there  can  be  no  part  of  the  graph  in  the 
second  section. 

If  -  2  <  X  <  1,  the  first  two  factors  are  positive  and  the  other  two  are  nega- 
tive ;  hence  the  radical  is  real  and  there  is  a  part  of  the  graph  in  the  third  section. 


124 


CERTAIN  ALGEBRAIC  FUNCTIONS 


If  1  <  X  <  4,  the  fii-st  three  factors  are  positive  and  the  last  is  negative  ; 
hence  the  radical  is  imaginary  and  there  can  be  no  part  of  the  graph  in  the 
fourth  section. 

Finally,  if  x  >  4,  all  the  factors  are  positive  ;  hence  the  i-adical  is  real  and 
there  is  a  part  of  the  graph  in  the  fifth  section. 

In  this  example  we  see  that  the  graph  consists  of  three  separate  parts,  and 
is  seen  (fig.  66)  to  consist  of  a  closed  loop  and  two  infinite  branches. 


Ex.3,    y  =  ±  V-  (X  +  4)  (X  +  2)  (X  -  1)  (X  -  4), 
The  plane  is  divided  into  five  sections  (fig.  67)  by  the  lines  x  = 
X  =  1,  and  X  =  4. 


-  4,  X  =  -  2, 


Fig.  67 


Proceeding   as   in   the  previous  two  examples,  we  find  y  to  be   real  if 

-4<x<-2,  or  l<x<4,  and  to  be  imaginaiy  for  all  other  values  of  x. 

Therefore  the  graph  consists  of  two  separate  parts,  and  is  seen  (fig.  67)  to 
consist  of  two  closed  loops. 


GKAPHS 


125 


66.  In  the  examples  of  the  last  article  no  two  factors  were 
alike,  i.e.  no  factor  occurred  more  than  once.  If  any  factor  does 
occur  more  than  once,  only  its  first  power  will  be  left  under  the 
radical  sign,  or,  to  put  it  more  generally,  no  perfect  square  will 
be  left  as  a  factor  under  the  radical  sign.  As  a  result,  there  will 
be  before  the  radical  a  factor  involving  x,  and  the  presence  of 
this  factor  will  of  necessity  change  the  course  of  the  reasoning  to 
some  extent,  as  is  shown  in  the  following  examples. 


Ex.  1.    y  =  ±  V(x  +  2)  (x  -  1)2. 
This  will  be  written  as 


y  =  ±  (X  -  1)  Vx  +  2. 


The  line  x  =  —  2  divides  the  plane 
(fig.  68)  into  two  sections. 

Proceeding  as  in  the  previous  ex- 
amples, we  find  the  radical  to  be  real 
if  X  >  —  2  and  imaginary  if  x  <  —  2. 
Therefore  there  is  a  part  of  the  graph 
to  the  right  of  the  line  x  =  —  2,  but 
there  can  be  no  part  of  the  graph  to 
the  left  of  that  line  unless  x  can  have 
such  value  as  to  make  the  coeflicient  of 
the  radical  zero ;  and  this  coefficient  is 
zero  only  when  x  equals  unity.  Hence 
all  of  the  graph  lies  to  the  right  of  the 
line  X  =  —  2,  as  shown  in  fig.  68. 

Comparing  this  example  with  Ex.  1 
of  §  65,  we  see  that  by  changing  the 
factor  X  —  5  to  x  —  1  we  have  joined 
the    infinite    branch     and     the    loop, 


making  a  single  curve  crossing  itself  at  the  point 


Ex.  2.    y  =  ±  V(x  +  2)2(x  -  1)  =  ±  (x  +  2)  Vx  -  1. 

The  line  x  =  1  divides  the  plane  (fig.  69)  into  two  sections. 

If  X  >  1,  the  radical  is  real  and  there  is  a  part  of  the  graph  in  the 
corresponding  section  of  the  plane.  If  x  <  1,  the  radical  is  imaginary  and 
there  will  be  no  points  of  the  graph  except  for  such  values  of  x  as  make 
the  coefficient  of  the  radical  zero.  There  is  but  one  such  value,  i.e.  -  2, 
and  therefore   there   is  but  one  point  of  the  graph,  i.e.  (—2,  0),   to  the 


126 


CERTAIN  ALGEBRAIC  FUNCTIONS 


left  of  the  line  x  =  1.     The  graph  consists,  then  (fig.  69),  of  the  isolated 
point  A  and  the  infinite  branch. 

Comparing  this  example  also  with  Ex.  1  of  §  65,  we  see  that  by  changing 
the  factor  x  —  5  to  x  +  2  we  have  reduced  the  loop  to  a  single  point,  leaving 
the  infinite  branch  as  such. 


Fig.  70 


Ex.3,    y  =  ±  V_  (X  +  4)  (X  +  2)2(x  -  4)  =  ±  (X  +  2)  V-  (X  +  4)  (X  -  4). 

The  lines  x  =  -  4  and  x  =  4  divide  the  plane  (fig.  70)  into  three  sections. 

If  -  4  <  X  <  4,  the  radical  is  real  and  there  is  a  part  of  the  graph  in  the  cor- 
responding portion  of  the  plane.  If  x  <  -  4  or  x  >  4,  the  radical  is  imaginary  ; 
and  since  in  the  corresponding  sections  there  is  no  value  of  x  which  makes  x  +  2 
zero,  there  can  be  no  part  of  the  graph  in  those  sections.-  It  is  represented  in 
fig.  70. 

Comparing  this  example  with  Ex.  3  of  §  65,  we  see  that  the  changingof  x  -  1 
to  X  +  2  has  brought  the  two  loops  together,  forming  a  single  closed  curve  cross- 
ing itself  at  the  point  ( -  2,  0). 


FUNCTIONS  DEFINED  BY  EQUATIONS 


127 


67.  Functions  defined  by  equations  of  the  second  degree  in  y. 

If  we  have  given  an  algebraic  equation  involving  both  y  and  x, 
y  is  thereby  defined  as  a  function  of  x.  For  if  x  is  assigned  any 
value,  the  corresponding  values  of  y  are  determined  by  means  of 
the  equation.  In  particular,  if  the  equation  involves  no  power  of 
y  higher  than  the  second,  it  may  be  readily  solved  for  y,  and  the 
work  of  finding  the  graph  is  similar  to  that  already  done. 

In  many  important  cases  the  solution  of  the  equation  is  of 
the  form 

y  =  c±  y/(x  —  Vj)  (x  —  r^)--'. 


Comparing  this  case  with  the  previous  one,  we  see  that  y  =  c 
is  an  axis  of  symmetry  instead  of  y  —  0,  and  that  in  all  other 
respects  the  work  is  similar. 


Ex.    2x2  +  2/2  + 3x -42/- 5  =  0. 
Solving  for  y,  we  have 


y  =  2±  V-2x2_  3x  +  9, 

or,  after  the  expression  under  the  radical 
sign  has  been  factored, 


y  =  2±  V-2(x-|)(x  +  3). 

The  lines  x  =  —  3  and  x  =  |  divide 
the  plane  (fig.  71)  into  three  sections,  and, 
proceeding  as  before,  we  find  that  the 
curve  is  entirely  in  the  middle  section, 
i.e.  when  —  3  <  x  <  ^,  and  that  the  line 
2/  =  2  is  an  axis  of  symmetry. 


Fig.  71 


In  case  the  given  equation  is 
of   higher  degree  in   y  than   the 

second,  but  of  the  first  or  the  second  degree  in  x,  it  is  evident 
that  we  can  solve  for  x  in  terms  of  y  and  proceed  as  above, 
working  from  the  y  axis  instead  of  the  x  axis. 

It  should  be  added  that  given  any  equation  in  x  and  y,  since 
either  may  be  regarded  as  the  independent  variable  and  the  other 
as  the  function,  we  have  perfect  freedom  of  choice  to  solve  for  y  in 
terms  of  x,  or  for  x  in  terms  of  y,  according  to  convenience. 


128 


CERTAIN  ALGEBRAIC  FUNCTIONS 


68.  Functions  involving  fractions.  If  the  expression  defining  a 
function  contains  fractions,  the  function  is  not  defined  for  a  value  of 
X  which  makes  the  denominator  of  any  fraction  zero  (§  11).  But  if 
a?  =  a  is  a  value  which  makes  the  denominator  zero,  but  not  the 
numerator,  and  x  is  allowed  to  approach  a  as  a  limit,  the  value  of 
the  function  increases  indefinitely  and  is  said  to  become  infinite.  The 
graph  of  a  fimction  then  runs  up  or  down  indefinitely,  approaching 
the  line  x  =  a  indefinitely  near,  but  never  reaching  it.  We  have 
thus  a  graphical  representation  of  the  discussion  of  infinity  in  §  11. 

When  a  fimction  becomes  infinite  it  is  discontinuous  (§  56). 
In  fact,  this  is  the  only  kind  of  discontinuity  which  can  occur  in 

an  algebraic  function. 


Ex.  1.    y 


Fig.  72 


a;-2 

It  is  evident  that  y  is 
real  for  all  values  of  x; 
also  if  a;  <  2,  y  is  negative, 
and  if  a;  >  2,  y  is  positive. 
Moreover,  as  x  increases 
toward  2,  y  is  negative 
and  becomes  indefinitely 
great ;  while  as  x  decreases 
toward  2,  y  is  positive  and 
becomes  indefinitely  great. 
We  can  accordingly  assign 
all  values  to  x  except  2, 
that  value  being  excluded 
by  §  11.  The  curve  is  repre- 
sented in  fig.  72. 

It  is  seen  that  the  nearer 
to  2  the  value  assigned  to  x, 
the  nearer  the  correspond- 
ing point  of  the  curve  to 
the  line  x  =  2.  In  fact, 
we  can  make  this  distance 
as  small  as  we  please  by 
choosing  an  appropriate 
value  for  x.    At  the  same 


time  the  point  recedes  indefinitely  from  OX  along  the  curve. 

Now  when  a  straight  line  has  such  a  position  with  respect  to  a  curve  that  as 
the  two  are  indefinitely  prolonged  the  distance  between  them  approaches  zero  as  a 
limit,  the  straight  line  is  called  an,  asymptote  of  the  curve. 


FUNCTIONS  INVOLVING  FRACTIONS 


129 


It  follows  from  the  above  definition  that  the  line  x  =  2  and  also  the  line 
y=0  are  asymptotes  of  this  curve.  In  this  example  it  is  to  be  noted  that  the 
asymptote  a;  =  2  is  determined  by  the  value  of  x  which  makes  the  function  infinite. 

It  is  clear  that  all  equations  of  the  type 
1 

y  = 

X  —  a 
represent  curves  of  the  same  gen- 
eral shape  as  that  plotted  in  fig.  72. 


Ex.2.    y  = 


+ 


x  +  2 

If  X  =  —  2  or  if  X  =  2,  y  is 
infinite ;  hence  these  two  values 
may  not  be  assigned  to  x,  all 
other  values,  however,  being 
possible.  The  curve  is  repre- 
sented in  fig.  73. 

By  a  discussion  similar  to  that 
of  Ex.  1,  it  may  be  proved  that 
the  lines  x  =  —  2  and  x  =  2, 
which  correspond  to  the  values 
of  X  which  make  the  function 
infinite,  and  also  the  line  y  =  0,  are  asymptotes  of  the  curve. 


This  curve  is  a  special  case  of  that  represented  by 


y 


1 


4- 


X  —  a      X  —  b 
and  it  is  not  difficult  to  see  how 
the  curve  represented  by 
1  1  1 


y 


+ 


+ 


+ 


X  —  a      X  —  b      X  —  c 
will  look  for  any  number  of  terms. 
1 


Ex.  3.    y  = 


(X  -  2)2 

All  values  of  x  may  be  assumed 
except  2.  The  curve  is  represented 
in  fig.  74.  It  is  evident  that  the  lines 
x  =  2  and  y  =  0  are  asymptotes. 

This  curve  is  a  special  case  of 
that  represented  by 
_        1 
^~(x-a)2' 
which  is  itself  a  special  case  of 
1  1 


Fig.  74 


(X  -  a)2      (X  -  6) 


+ 


130 


CERTAIN  ALGEBRAIC  FUNCTIONS 


Ex.  4.    y*  = 


x-3 


As  in  §  67,  we  solve  for  y,  forming  tlie  equation 


ion  y  =  ±yj^ 


The  line 


X  =  3  (fig.  75)  divides  the  plane  into  two  sections,  and  it  is  evident  that  there 
can  be  no  part  of  the  curve  in  that  section  for  which  a;  <  3.    Moreover,  this 


Fig.  75 

line  X  =  3  is  an  asymptote,  as  in  the  preceding  examples.    The  curve,  which 
is  a  special  case  of  that  represented  by 


Is  represented  in  fig.  75.   It  is  to  be  noted  that  the  axis  of  x  also  is  an  asymptote. 
Ex.5.    y  =  ?i±i. 

X 

To  plot  this  curve  we  write  the  equation  in  the  equivalent  form 


y  =  x  +  -. 


(1) 

It  is  evident  that  all  values  except  0  may  be  assigned  to  x,  that  value  being 
excluded  as  it  makes  y  infinite.    Let  us  also  draw  the  line 


y  =  x, 


(2) 


a  straight  line  passing  through  the  origin  and  bisecting  the  first  and  the  third 
quadrants. 


SPECIAL  IRRATIONAL  FUNCTIONS 


13i 


Comparing  equations  (1)  and 
(2),  we  see  that  if  any  value  xj, 
is  assigned  to  x,  the  corre- 
sponding ordinates  of   (1)  and 

(2)  are  respectively  xi  -\ and 

Xi     ■, 

Xi,  and  that  they  differ  by  — . 

Moreover,  the  numerical  value 
of  this  difference  decreases  as 
greater  numerical  values  are 
assigned  to  Xi,  and  it  can  be 
made  less  than  any  assigned 
quantity  however  small  by  tak- 
ing Xi  sufficiently  great.  It 
follows  that  the  line  y  =  x  isan 
asymptote  of  the  curve.  It  is 
also  evident  that  the  line  x  =  0, 
determined  by  the  value  of  x 
which  makes  the  function  infi- 
nite, is  an  asymptote.  The  curve 
is  represented  in  fig.  76. 


(£)A 


Fig.  70 


69.  Special  irrational  functions. 

Ex.  1.    2/2  =  x3. 

Writing  this  equation  in  the  form  y  =  ±x  Vx, 
we  see  that  y  is  an  irrational  function  of  x,  and 
that  its  graph  is  symmetrical  with  respect  to 
OX  and  lies  entirely  to  the  right  of  the  axis  y. 
It  is  represented  in  fig.  77,  and  is  called  the 
semicubical  parabola. 

In  general,  if  the  equation  expressing  the 
_y  function  is  of  the  form 

y  =  kx", 

the  function  is  rational  or  irrational  according 
as  n  is  integral  or  fractional.  In  §  38  we  have 
plotted  the  graphs  of  some  of  the  rational  func- 
tions of  this  type  for  the  special  case  when  k=  1 
and  n  has  the  values  3,  4,  and  5  respectively. 
Above  we  have  just  plotted  the  graph  of  one  of 
the  irrational  functions,  i.e.  when  n  =  ^. 

The  grajDhs  of  the  irrational  functions  y  =  x^, 

■pio,  77  y  =  X*,  and  y  =  x^  may  be  obtained  by  assuming 

values  for  x  and  plotting  as  above,  or  by  rewriting 

the  equations  in  the  forms  x  =  y^,  x  =  y*,  and  x  =  y^,  when  it  is  immediately 

evident  that  their  graphs  are  respectively  the  same  in  shape  as  those  of  the 


132 


CEKTAIN  ALCiEBRAIC  FUNCTIONS 


Fig.  78 


rational  functions  y  =  x^,  y  =  x*,  and 
y  =  x^  already  plotted,  the  axes  of 
X  and  y,  however,  being  changed  in 
position. 

It  is  to  be  noted  that  the  graphs  of 
all  the  functions  expressed  by  the 
equation  y  =  x"  pass  through  the  points 
(0,  0)  and  (1,  1). 

Ex.  2.  a;'  +  2/5  =  a'. 
If  y  is  defined  as  a  function  of  x 
the  equation  x'  +  y^  =  a\  it  is 
evident  that  its  graph  will  lie  entirely 
in  the  first  quadrant,  since  both  x 
and  y  must  be  positive,  and  that  its  relative  positions  with  respect  to  the 
two  axes  of  coordinates  are  the  same 
(fig.  78).  The  curve  is  a  parabola 
(§  79).  If  the  equation  is  put  in  the 
form  2/  =  (a*  —  x^)-,  it  is  .seen  that  y 
is  an  irrational  function  of  x. 

Ex.  3.    xi  +  y^  =  al 

Writing  this  equation  in  the  form 
y  =  ±  (a*  —  x^)-,  we  see  that  2/  is  an 
irrational  function  of  x,  and  that  its 
graph  is  symmetrical  with  respect  to 
OX  and  bounded  by  the  lines  x  =  —  o 
and  X  =  a.  In  the  same  way  we  may 
show  that  the  graph  is  symmetrical 
with  respect  to  OY  and  bounded  by  the 
lines  y  =  ~  a  and  y=  a.  It  is  repre- 
sented in  fig.  79,  and  is  a  four-cu.sped  hypocycloid. 

Ex.  4.    x3  +  y^  -3axy  =  0. 

The  graph  of  this  equation, 
by  which  y  is  defined  as  an 
irrational  function  of  x,  is  repre- 
sented in  fig.  80,  and  is  known 
as  the  Folium  of  Descartes. 
It  is  symmetrical  with  respect 
to  the  line  y  =  x  and  has  the 
line  X  +  y  +  a  =  0  as  an 
asymptote.  While  it  may  be 
plotted  by  assuming  values  for 
X  and  solving  the  corresponding 
cubic  equations  for  y,  it  is  more 
easily  plotted  when  different 
axes  of  coordinates  are  chosen 
(see  Ex.  38,  Chap.  X). 


PROBLEMS 


.00 


PROBLEMS 
Plot  the  graphs  of  the  following  equations : 

1.  y2  =  (X  -  1)  {x'^  -  4).  29.  2/3  =  a;2(x  +  2). 

2.  y2  ^  (X  +  2) (8x  -  x2  _  15).  30.   {y  +  2)3  =  (x  -  1) (x2  -  4). 

3.  4  2/2  =  (x  +  3)(2x-3)2.  31.  X2/  =  7. 

4.  42/2  =  x2{x  +  l).  •  32.  xy  =  -1. 

5.  2/2  =  (X- 3)2 (.5-2 X).  __  16 

66.  y  = 

6.  ?/2  =  (3x  +  2)(9x2-4).  ^-* 

7.  2/2  =  (X  -  2)2(4x2  -  4x  -  15).  34.  ?/  = ^- L_. 

(X  -  1)2       (X  +  3)2 

8.  y2^(4x2-l){x2-4). 

9.  2/2  =  {2x+5)2(6  +  x-x2).  ^^-  22/  =  3x4--- 

10.  2/2  =  -x2(x  +  3)2(x  +  l).  36.  2/-2  =  2(x-l)+     ^     . 

11.  2/2  =  x2{x-2)2(x-3).  „„  1 

37.  (2/ -2)2  =  ^. 

12.  2/2  =  (1  -  x2)  (x2  -  9).  a^  +  1 

13.  2/2  =  (2x-5)(x2  +  2).  38.  y2  =  5i^±^. 

14.  2/2=(x-2)2(x2-f-2). 

15.  2/-  =  (x-2)(2x-3)2(x2+x+l).  ^^    y^^^Zr^- 

16.  10^2-  42-4  _a;6.  3 

40.  2/2  = ? 

17.  x2-2/2-4x  +  6y-l  =  0.  x2-6x  +  8 

18.  4x2  +  92/2+ 4x- 122/ -31=0.  *^-  a;22/2  +  36  =  4 2/2. 

19.  x2  -  2/3  +  32/2  +  2/  -  3  =  0.  ^2-  l^'^'^/'  =  62a;2(a2  -  2ax). 

20.  x2-2/4(4  +  2/)  =  0.  43.  2/2  =  ^!^^^±^. 

21.  [x2  +  .S(2/-l)][x2-3(2/-l)]=0. 

44.  2/(x2  +  a2)  =  a2(a-x). 

22.  (2/-l)2  =  (x-l)2(x-4). 

45.  2/2(x2  +  a2)_a2a;2. 

23.  (2/ _  x)2  =  9  -  x2. 

46.  a*2/2  +  62a;4  =  a262x2. 

24.  (X  +  ?/)2=:  2/2(2/  +  1). 

^         ^       •^Vi'^;  47.  2/2(a2  +  x2)  =  x2(a2-x2). 

25.  x2  -  4  X2/  +  8  2/2  -  2/*  =  0. 

48.  xy2  =  4a2(2a-x). 

^^■^h^h'-  49..  =  x=  +  i. 

27.  2/3  =  x4.  J 

28.  2/3  =  X (x2  -  4).  •  ^  ^  ^  +  ^" 


CHAPTEE  VII 


CERTAIN  CURVES  AND  THEIR  EQUATIONS 


70.  The  circle.  Wheu  a  curve  has  been  defined  by  a  geometric 
property  it  is  often  possible  to  find  the  equation  of  the  curve  by 
expressing  the  definition  in  algebraic  symbols.  This  equation 
serves,  then,  as  a  means  for  plotting  the  curve  and  also  as  a  basis 
for  examiniug  its  other  properties.  In  this  chapter  we  shall  derive 
the  equation  of  certain  important  elementary  curves,  beginning 
with  the  circle. 

A  circle  is  the  locus  of  a  point  at  a  constant  distance  from  a 
fixed  point.    The  fixed  point  is  the  center  of  the  circle  and  the 

constant  distance  is  the  radius. 

Let  {d,  e)  (fig.  81)  be  the  coordi- 
nates of  the  center  C,  and  r  the 
radius  of  the  circle.  Then  if  P  (x,  y) 
is  a  point  on  the  circle,  x  and  y 
must  satisfy  the  equation 

(x-df^{y-ef  =  r',       (1) 

by  §  17. 

Conversely,  if  x  and  y  satisfy 
the  equation  (1),  the  point  {x,  y) 
is  at  a  distance  r  from  {d,  e)  and 
therefore  lies  on  the  circle. 

Therefore  (1)  is  the  equation  of  the  circle  (§  22). 

Equation  (1)  expanded  gives 

x-+f~  2dx-2ey  +  d'+e^-r^=0; 
and  if  this  is  multiplied  by  any  quantity  A,  it  becomes 

Ax'  +  Af+2Gx+2Fy  +  C=0, 
where 


Fig.  81 


(2) 


d=-- 
A 


e=--,    d?Jre^-r'  =  -' 
A  A 

134 


THE  CIRCLE  135 

Ex.    The  equation  of  a  circle  with  the  center  (^,  —  J)  and  the  radius  §  is 
(a;-J)2  +  (y  +  J)2  =  4, 
which  reduces  to        12  a;^  +  12  ys  _  12  x  +  8  y  -  1  =  0. 

71.  Conversely,  the  equation 

where  A^  0,  represents  a  circle,  if  it  represents  any  curve  at  all. 
To  prove  this,  we  will  transform  the  equation  as  follows : 

a?+2-x  +  y^  +  2-y=--, 
A         ^  A"^  A 

^,0^      ,  <^\     2,0^      ,  ^'      G^+F^-AC 


4/    y    A I  A' 

There  are  then  three  possible  cases : 

1.  G^  +  F^-AOO.    The  equation  is  then  of  the  type  (1),  §  70, 

1,       ^         ^  ^'    -2     G^+F^~AC       ,^,       , 

where  a  = 7'^  = ,r=^ — ,  and  therefore  represents 

A  A  A^  . 

a  circle  with  the  center  ( , |  and  the  radius  \ 

\     A        AJ  >  A^ 

2.  G''  +  F--AC=Q.    The  equation  is  then 

which   can  be  satisfied  by  real  values   of  x  and  y  only  when 

G  F 

x  = and  y— Hence  the  equation  represents  the  point 

— -> I .    This  may  be  called  a  circle  of  zero  radius,  regarding 

it  as  the  limit  of  a  circle  as  the  radius  approaches  zero. 

3.  G^-\-F^  —  AC<0.  The  equation  can  then  be  satisfied  by  no 
real  values  of  x  and  y,  since  the  sum  of  two  positive  quantities 
cannot  be  negative.    Hence  the  equation  represents  no  curve. 


136      CERTAIN  CURVES  AND  THEIR  EQUATIONS 

Ex.  1.    The  equation  x2  +  j/2-2x  +  4y4-l  =  0  may  be  written 
(X  -  1)2  +  (2/  +  2)2  =  4, 
and  represents  a  circle  with  center  (1,  -  2)  and  radius  2. 

Kx.  2.    The  equation  x^  +  y-  -  2 x  +  iy  +  o  =  0  may  be  written 
(x'- 1)2 +  (y  + 2)2  =  0, 
and  is  satisfied  only  by  the  point  (1,  -  2). 

Ex.  3.    The  equation  x^  +  y^  -2x  +  4y  +  7  =  0  may  be  written 
(X  -  1)2  +  (y  +  2)2  =  -  2, 
and  represents  no  curve. 

72.  To  find  the  equation  of  a  circle  which  will  satisfy  given 
conditions,  it  is  necessary  and  sufficient  to  determine  the  three 
quantities  d,  e,  r,  or  the  ratios  of  the  four  quantities  A,G,F,  C. 
Each  condition  imposed  upon  the  circle  leads  usually  to  an  equa- 
tion involving  these  quantities.  In  order  to  determine  the  three 
quantities  it  is  necessary  and  in  general  sufficient  to  have  three 
equations.  Hence,  m  general,  three  conditions  are  necessary  and 
sufficient  to  determine  a  circle. 

It  is  not  important  to  enumerate  all  possible  conditions  which 
may  be  imposed  upon  a  circle,  but  the  following  three  may  be 
mentioned. 

1.  Let  the  condition  be  imposed  upon  the  circle  to  pass  through 
the  known  point  (x^,  y^.  Then  {x^,  y^  must  satisfy  the  equation 
of  the  circle ;  therefore  d,  e,  and  r  must  satisfy  the  condition 

{x,-df+{y-ef  =  r\ 

2.  Let  the  condition  be  imposed  upon  the  circle  to  be  tangent 
to  the  known  straight  line  Ax  +  By  +  C=Q.  Then  the  distance 
from  the  center  of  the  circle  to  this  line  must  equal  the  radius ; 
therefore,  by  §  32,  d,  e,  and  r  must  satisfy  the  condition 

Ad  +  Be  +  C 

-  =  ±  r. 

y/A'  +  B^ 

The  sign  will  be  ambiguous,  unless  from  other  conditions  of  the 
problem  it  is  known  on  which  side  of  the  line  the  center  lies. 


THE  CIRCLE  137 

3.  Let  it  be  required  that  the  center  of  the  circle  should  lie  on 
the  line  Ax  +  i?y  +  C=  0,    Then  d  and  e  must  satisfy  the  condition 

Ad  +  Be  +  C=Q. 

Ex.  1.    Find  the  equation  of  the  circle  through  the  three  points  (2,  —  2), 
(7,  3),  and  (6,  0). 

The  quantities  d,  e,  and  r  must  satisfy  the  three  conditions 

(2-d)2  +  (-2-e)2  =  r2, 
(7  -  d)2  +  (3  -  e)2  =  r2, 
(6  -  d)2  +  (0  -  e)2  =  r2. 

Solving  these  we  have  d  =  2,  e  =  3,  and  r  =  5.     Therefore  the   required 
equation  is 

(x-2)2  +  (y._3)2  =  25, 

or  x2  +  2/2  -  4  a;  -  6  y  -  12  =  0. 

Ex.  2.    Find  the  equation  of  the  circle  which  passes  through  the  points 
(2,   —3)   and   (—4,   —1)   and   has   its   center   on   the   line   Sy  +  x  —  IS  =  0. 
The  quantities  d,  e,  and  r  must  satisfy  the  conditions 

(2-d)2+ (-3-e)2  =  r2, 

(_4_d)2  +  (_l_  e)2  =  r-2, 

3  e  +  d  -  18  =  0. 

Solving  these  equations  we  find  d  =  |,  e  =  y-,  r2=-l-|-S..     Therefore  the 
required  equation  is 

or  a;2  +  ^2  _  3a;  _  11 2/  -  40  =  0. 

Ex.  3.    Find  the  equation  of  a  circle  which  is  tangent  to  the  lines 

17  x  + 2/ -35  =  0     and     13x  +  II2/ +  50  =  0, 

and  has  its  center  on  the  line  88  x  +  70  2/  +  15  =  0. 

The  quantities  d,  e,  and  r  must  satisfy  the  conditions 

17d  +  e-35 

■■±r. 


V290 
-  13d -lie -50 


=  ±r, 


V290 
88d  +  70e  +  15  =  0. 


138      CERTAIN  CURVES  AND  THEIR  EQUATIONS 

These  equations  have  the  two  solutions 

and  d  =  6,  e  =  —  ^^ , 


6      ' 
3V29O 


20 

Hence  each  of  the  two  circles 

3a;2  +  32/2+  5x-  5j/-20  =  0 
and  40x2  +  402/2  _400x  + 620y  + 2429  =  0 

satisfies  the  conditions  of  the  problem. 


Ex.  4.    The  equation  of  a  circle  through  three  given  points  is  most  readily 
found  by  means  of  the  equation 

^x2  +  .42/2  +  2Gx  +  2Fy-\-C=0. 

If  (a^i,  yi),  {3^1  1/2)1  and  (X3,  2/3)  are  the  three  given  points,  the  quantities 
A,  G,  F,  C  must  satisfy  the  equations 

^x  2  +  Ay^  +  2Gxi  +  2Fyi  +  C=0, 
Ax^  +  Ayl  +  2  Gx2  +  2  Fyo  +  C  =  0, 
Ax^  +  Ay^  +  2  Gx3  +  2  Fys  +  C  =  0. 

There  are  here  four  homogeneous  equations  in  the  unknowns  .4,  G,  jP,  C, 
and  the  result  of  eliminating  the  unknowns  is,  by  §  9, 


x2  +  ?/2  X  y  1 

^X+Vl  *!  Vi  1 

xi  +  yi  X2  2/2  1 

^s+v!  X3  Vs  1 


0, 


(1) 


which  is  the  required  equation  of  the  circle. 

It  is  to  be  noticed  that  the  coefficient  of  x2  4-  2/2  in  (1)  is 


xi    Vi     11 
X2    2/2     1 1 . 
xs    Vz     11 


When  this  is  zero,  equation  (1)  is  of  the  first  degree  and  represents  a  straight 
line.    But  when 


Xi 

2/1      1 

X2 

Vi     1 

X8 

2/8      1 

=  0, 


the  points  (Xi,  2/1),  (X2, 2/2),  and  (X3, 2/3)  are  on  the  same  straight  line  (§  29,  5)  and 
cannot  determine  a  circle. 


THE  ELLIPSE 


139 


73.  The  ellipse.  A71  ellipse  is  the  locus  of  a  point  the  sum  of 
the  distances  of  which  from  two  fixed  points  is  constant. 

The  two  fixed  points  are  called  the  foci.  Let  them  be  denoted 
by  F  and  F'  (fig.  82)  and  let  the  axis  of  x  be  taken  through  them 
and  the  origin  halfway  between  them.  Then  if  P  is  any  point 
on  the  ellipse  and  2  a  represents  the  constant  sum  of  its  distances 
from  the  foci,  we  have 

F'P  +  FP=2a.  (1) 

From  the  triangle  F'PF  it  follows  that 

F'F<2a. 

Hence  there  is  a  point  A  on  the  axis  of  x  and  to  the  right 
of  F  which  satisfies  the  definition.    We  have  then 


F'A  +  FA=2a, 
or  {F'O  +  OA) + {OA-  OF)  =2  a, 
whence  OA  =  a. 

Let  us  now  place 
OF 


OA 


=  e,   where    e  <  1. 


Then  the  coordinates  of  F  and 
F'  are  {±ae,  0).  Computing 
the  values  of  F'P  and  FP  by  §  17,  and  substituting  in  (1),  we  have 


V(a?+  aeY+  y^  +  V(^  —  ae)'^  +  y^  =  2> 


(2) 


By  transposing  the  second  radical  to  the  right-hand  side  of  the 
equation,  squaring,  and  reducing,  we  have 


a  —  ex=  ^{x—aeY+f  =  FP.  (3) 

Similarly,  by  transposing  the  first  radical  in  (2),  we  have 

a  +  ex  =  y/{x  +  aef  +f  =  F'P.  (4) 


140      CERTAIN  CURVES  AND  THEIR  EQUATIONS 
Either  (3)  or  (4)  leads  to  the  equation 

or  -,+    ./    ,  =1.  (6) 

Since  e  <1,  the  denominator  of  the  second  fraction  is  positive 
and  we  place  „  ,       ,o 

thus  obtaining  ~2  +  ^2  ~  -^-  C^) 

We  have  now  shown  that  any  point  which  satisfies  (1)  has  co- 
ordinates which  satisfy  (7). 

To  show,  conversely,  that  any  point  whose  coordinates  satisfy  (7) 
is  such  as  to  satisfy  (1),  let  us  assume  (7)  as  given.  We  can  then 
obtain  (6)  and  (5),  and  (5)  may  be  put  in  each  of  the  two  forms 

ar^  +  2  aex  +  a^c^  +  if  =  «-  +  2  aex  +  ^o?, 
ar^  —  2  aex  +  cj^c^  +  if  =  ar—1  aex  +  e^x?, 

the  square  roots  of  which  are  respectively 

F'F=±{a  +  ex), 
FP  =  ±{a  —  ex). 

These  lead  to  one  of  the  four  following  equations : 

F'P+FP=2a, 

F'P-FP=2a, 
-F'P  +  FP=2a, 
-F'P-FP  =  2a. 

Of  these,  the  last  one  is  impossible,  since  the  sum  of  two  nega- 
tive numbers  cannot  be  positive;  and  the  second  and  third  are 
impossible,  since  the  difference  between  FP  and  F'P  must  be  less 
than  F'F,  which  is  less  than  2  a.  Hence  any  point  which  satisfies 
(7)  satisfies  (1),  and  therefore  (7)  is  the  equation  of  the  ellipse. 


THE  ELLIPSE 


141 


74.  Placing  y  =  0  in  (7),  §  73,  we  find  x  =  ±  a.  Placing  x  =  0, 
we  find  y  =  ±h.  Hence  the  ellipse  intersects  OX  in  the  two  points 
A{a,  0)  and  A' {—  a,  0),  and  intersects  OF  in  two  points  B{Q,  h) 
and  -B'(0,  —  h).  The  points  A  and  ^'  are  called  the  vertices  of  the 
ellipse.  The  line  AA',  which  is  equal  to  2  a,  is  called  the  mayor 
axis,  and  the  line  ^^',  which  is  equal  to  2  h,  is  called  the  wtTior 
axis  of  the  ellipse. 

Solving  (7)   first  for   y  and 
then  for  x,  we  have 


M 


y  =  ±-^a'-ci» 


A' 


and 


a;  =  ±  -  V&-^  -  y"^ 


B' 
Fig.  83 


iV  B'  K 

These  equations  show  (1)  that 
the  elHpse  is  symmetrical  with 
respect  to  both  OX  and  OY,  (2)  that  x  can  have  no  value  numer- 
ically greater  than  a,  (3)  that  y  can  have  no  value  numerically 
greater  than  b.  If  we  construct  the  rectangle  KLMN  (fig.  83), 
which  has  0  for  a  center  and  sides  equal  to  2  a  and  2  h  respec- 
tively, the  ellipse  will  lie  entirely  within  it ;  and  if  the  curve  is 
constructed  in  one  quadrant,  it  can  be  found  by  symmetry  in  all 
quadrants.    The  form  of  the  curve  is  shown  in  figs.  82  and  83. 

75.  Any  equation  of  the  form  (7),  §  73,  in  which  a  >  b, 
represents  an  ellipse  with  the  foci  on  OX.  For  if  we  place, 
as  in  §73,  &"' =  a'(I-e'),  we  find 


Vc^^' 


and  may  fix  i^and  F',  which  in  §  73  were  arbitrary  in  position,  by 
the  relation  0F  =  —  OF'  =  ae. 

The  foci  may  be  found  gi'aphically  by  placing  the  point  of  a  com- 
pass on  B  and  describing  an  arc  with  the  radius  a.  This  arc  will 
intersect  AA'  in  the  foci ;  for  since  OB  =  h  and  OF  =  Va^  —  b\ 
BF=a. 


142      CERTAIN  CURVES  AND  THEIR  EQUATIONS 

Similarly  an  equation  of  the  form  (7),  §  73,  in  wliich  h>a, 
represents  an  ellipse  in  which  the  foci  lie  on  BB'  at  a  distance 
VV^—a^  from  0.  In  this  case  BB'  =  2  &  is  the  major  axis  and 
AA'  =  2  a  is  the  minor  axis. 

It  may  be  noted  that  the  nearer  the  foci  are  taken  together,  the 
smaDer  is  e  and  the  more  nearly  h  =  a.  Hence  a  circle  may  he 
considered  as  an  ellipse  with  coincident  foci  and  equal  axes. 

76.  The  hyperbola.  An  hyperhola  is  the  locus  of  a  point  the 
difference  of  the  distances  of  which  from  two  fixed  points  is  constant. 


Fig.  84 

The  two  fixed  points  are  called  the  foci.  Let  them  be  F  and 
F'  (fig.  84)  and  let  FF'  be  taken  as  the  axis  Of  x,  the  origin  being 
lialfway  between  F  and  F'.  Then  if  P  is  any  point  on  the 
hyperbola  and  2  a  is  the  constant  difference  of  its  distances 
from  F  and  F',  we  have  either 


or 


F'P-FP  =  2a, 
FP-F'P  =  2a. 


(1) 
(2) 


Since  in  the  triangle  F'PF  the  difference  of  the  two  sides  FP 
and  F'P  is  less  than  F'F,  it  follows  that  F'F  >2a. 

There  is  therefore  at  least  one  point  A  between  O  and  F  which 
satisfies  the  definition. 


THE  HYPERBOLA  143 

Then  F'A—AF=2a, 

or  {F'0  +  0A)-{0F-0A)=2a; 

whence  OA  =  a. 

We  may  therefore  place 

— -  =  e,  where  e  >  1. 
OA 

Then  the  coordinates  of  F  and  F'  are  (±  ae,  0)  and  equations 
(1)  and  (2)  become 


V(aj  +  aef  +  /  —  V(a;  —  ae)"-^  +f  =  2a  (3) 

and  V(a;  —  aef  +  2/"^  —  V(a;  +  aef  +  y-  =  2  a.  (4) 

By  transposing  one  of  the  radicals  to  the  right-hand  side  of 
these  equations,  squaring,  and  reducing,  we  obtain  from  (3)  either 


+  a=  y/{x  +  aef  +  f  =  F'P, 
—  a  =  y/{x  —  aef  -\-f=FP', 
and  from  (4)  we  obtain  either 


ex 
or  ex 


■  {ex  +  a)  =  y/{x  +  aef+y''  =  F'P, 


or  —  {ex  —  a)  =  ■>J{x  —  aef  +  /  =  FP. 

Any  one  of  the  last  four  equations  gives 


{l-e')x'+f  =  d\l-e\  (5) 

^  ,  f 

a^      a\l-e^) 


^.+  ..f    ..=!•  (6) 


But  since  e  >l,a^{l  —  e'^)  is  a  negative  quantity  and  we  may 
write  a^{l  —  e^)  =  —  b^,  thus  obtaining 

^-C  =  l.  (7) 


b 


144      CERTAIN  CURVES  AND  THEIR  EQUATIONS 

Then  any  point  wliich  satisfies  (1)  or  (2)  satisfies  (7).  Conversely, 
by  retracing  our  steps,  we  find  that  if  the  coordinates  of  a  point 
P  satisfy  (7),  then 

F'P  =  ±(ex  +  a) 
and  FP  =  ±{ex  —  a). 

Hence  we  must  have  either 

F'P  —  FP  =  2a, 

—  F'P+FP  =  2a, 

F'P  +  FP  =  2a, 

or  -F'P  —  FP  =  2a. 

The  equation  F'P  +  FP  =  2  a  is  impossible,  for  F'P  +  FP>  F'F, 
and  2a  < F'F.  The  equation  —  i^'P  —  i^P  =  2 a  is  also  clearly 
impossible.  Hence  any  point  which  satisfies  (7)  satisfies  either 
(1)  or  (2).    Therefore  (7)  is  the  equation  of  the  hyperbola. 

77.  If  we  place  y  =  0  in  (7),  §  76,  we  have  x  =  ±a.  Hence 
the  curve  intersects  OX  in  two  points,  A  and  A',  called  the  vertices. 
It  x=  0,  y  is  imaginary.    Hence  the  curve  does  not  intersect  OY. 

Solving  (7),  §  76,  for  y  and  x  respectively,  we  have 


a 


and  x=±-  Vy+i^. 


These  show  (1)  that  the  curve  is  symmetrical  with  respect  to 
both  OX  and  OY,  (2)  that  x  can  have  no  value  numerically  less 
than  a,  and  (3)  that  y  can  have  all  values. 

Moreover,  the  equation  for  y  can  be  written 


THE  HYPERBOLA 


145 


As  X  increases  the  term  —  decreases,  approaching  zero  as  a  limit. 
Hence  the  more  the  hyperbola  is  prolonged,  the  nearer  it  comes 
to  the  straight  lines  y  =  ±  -  x.  Therefore  the  straight  lines 
y  =  ±-  X  are  the  asymptotes  of  the  hyperbola.  They  are  the 
diagonals  of  the  rectangle  constructed  as  in  fig.  85,  and  are  used 

F 


Fig.  85  , 

conveniently  as  guides  in  drawing  the  curve.     The  line  AA'  is 

called  the  transverse  axis  and  the  hne  BB'  the  conjugate  axis 

of  the  hyperbola.     The    shape   of   the   curve  is  shown  in   figs. 

84  and  85. 

78.  Any  equation  of  the  form  (7),  §  76,  where  a  and  h  are  any 

positive  real  values,  represents  an  hyperbola  with  the  foci  on  AA'. 

VaF+l? 
For  if  we  place  —h^  =  a'^(l  —  e'),  we  find  e  =  ■ and  may 


146      CERTAIN  CURVES  AND  THEIR  EQUATIONS 


find  the  position  of  the  foci  from  the  equations  OF  =  —  OF'  =  ae. 
Similarly  any  equation  of  the  form 

represents  an  hyperbola  mth  the  foci  on  BB'. 

\ih  =  a,  the  hyperbola  is  called  an  equilateral  hyperhola  and  its 
equation  is  either  ar^—  z/^  =  a^  or  —  ar^+  y^  =  a^ 

79.  The  parabola.  A  parabola  is  the  locus  of  a  point  equally 
distant  from  a  fixed  point  and  a  fixed  straight  line.  The  fixed 
point  is  called  the  focus  and  the  fixed  straight  line  the  directrix. 
Let  the  line  through  the  focus  perpendicular  to  the  directrix  be 
taken  as  the  axis  of  x,  and  let  the  origin  be  taken  on  this  line  halfway 

between  the  focus  and  the  directrix. 
*  ^  Let  us  denote  the  abscissa  of  the 

focus  by  p.  In  fig.  86  let  i^  be  the 
focus,  BS  the  directrix  intersecting 
OX  at  D,  and  let  P  be  any  point  on 
the  curve.  Then  the  coordinates  of 
-X  F  are  (p,  0),  those  of  D  are  {—p,  0), 
and  the  equation  of  BS  is  «  =  —p. 
Draw  from  P  a  line  parallel  to  OX 
intersecting  BS  in  N.  If  F  is  on  the 
right  of  BS,  P  must  also  lie  on  the 
right  of  BS,  and  by  the  definition 

FP  =  NP. 


If,  on  the  other  hand,  F  is  on  the  left  of  BS,  P  is  also  on  the 
left  of  BS  and 

FP  =  PN=-NP. 

In  either  case  FP^  =  NP^. 


But  FP  =  (x-pY-\-  f,  and  NP  =  x  +  p; 

hence  {x~pf+f  =  (x  +  p)', 

which  reduces  to  -f  =  i^px. 


(by  §  17) 

(1) 


THE  PARABOLA 


147 


Any  point  on  the  parabola  then  satisfies  this  equation. 
Conversely,  it  is  easy  to  show  that  if  a  point  satisfies  this 
equation,  it  must  so  lie  that  FP  =  ±  NP,  and  hence  lies  on 
the  parabola. 

Equation  (1)  shows  (1)  that  the  curve  is  symmetrical  with 
respect  to  OX,  (2)  that  x  must  have  the  same  sign  as  p,  and  (3) 
that  y  increases  as  x  increases  numerically.  The  position  of  the 
curve  is  as  shown  in  fig.  86  when  jp  is  positive.  When  p  is  neg- 
ative F  lies  at  the  left  of  0  and  the  curve  extends  toward  the 
negative  end  of  the  axis  of  x. 

Similarly  the  equation  a?  =  4:py  represents  a  parabola  for  which 
the  focus  lies  on  the  axis  of  y,  and  which  extends  toward  the 
positive  or  the  negative  end  of  the  axis  of  y  according  as  p  is 
positive  or  negative.  In  all  cases  0  is  called  the  vertex  of  the 
parabola  and  the  line  determined  by  0  and  F  is  called  its  axis. 

80.  If  Pi{x^,  2/i)  and  Po{x^,  y„)  are  two  points  on  the  parabola 
y^  =  4:px  (fig.  87),  then 


yl  =  ^px^; 


hence 


Fig.  87 


That  is,  tlie  squares  of  the  ordinates 

of  a  parabola  are  to  each  other  as  the 

abscissas.    Conversely,  if  in  any  curve  the  squares  of  the  ordinates 

are  to  each  other  as  the  abscissas,  the  curve  is  a  parabola. 

For  let  i^  be  a  known  point  and  P  any  point  on  the  curve. 
Then,  by  hypothesis. 


which  may  be  written 


y^  =  ^x. 


But  this  is  the  same  as  y^  =  4:px,  where  p  = 


148      CERTAIN  CURVES  JlSD  THEIR  EQUATIONS 


81.  The  conic.  A  conic  is  the  locus  of  a  point  the  distance 
of  which  from  a  Jixed  point  is  in  a  constant  ratio  to  its  distance 
from  a  fixed  straight  line. 

The  fixed  point  is  called  the  focus,  the  fixed  line  the  directrix, 
and  the  constant  ratio  the  eccentricity. 

We  shall  take  the  directrix  as  the  axis  of  y,  and  a  line  through 
the  focus  F  as  the  axis  of  x,  and  shall  caU  the  coordinates  of  the 

focus  (c,  0),  where  c  represents  OF  and 
is  positive  or  negative  according  as  F 
lies  to  the  right  or  the  left  of  O. 

Let  P  be  any  point  on  the  conic; 
connect  P  and  F,  and  draw  PN  per- 
pendicular to  OY.    Then  by  definition 

FP  =  ±eXP,  (1) 

according  as  P  is  on  the  right  or  the 
left  of  OT.    In  both  cases 

FP*  =  e'-  XP\ 

But  FP*  =  (a: — ef  +  y»,  by  §  17,  and 
NP  =  z.  Therefore  for  any  point  on 
the  conic 


r 

/ 

/ 
/•I 

/ 

0 

\ 

\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
\ 

Fio.  88 


{x-c)'+f  =  t^Ji*. 


(2) 


It  is  easy  to  show,  conversely,  that  if  the  coordinates  of  P  sat- 
isfy (2),  P  satisfies  (1).    Hence  (2)  is  the  equation  of  the  conic. 

It  is  clear  that  the  parabola  is  a  special  case  of  a  conic,  for  the 
definition  of  the  latter  becomes  that  of  the  former  when  e  =  1. 

It  is  also  not  difficult  to  show  that  the  ellipse  is  a  special  case 
of  a  conic,  where  the  eccentricity  is  «  of  §  73  and  <  1. 

For  if  P  (fig.  89)  is  a  point  on  the  ellipse  -i  +  ^  =  1>  we  found 
in  §  73  that  a       6- 

FP  =  a-  ex,  F'P  =  a  +  ex, 

or  FP  =  e(^-x\        F'P  =  e(-  +  x\ 


THE  CONIC 


149 


If  now  we  take  the  point  D  so  that  OD  —  - »  and  Z)'  so  that 


OD'  = »  draw  the  lines  DS  and  Z>'*Sf'  perpendicular  to  OX,  the 

line  N'FN  perpendicular  to  I>S,  and  the  ordinate  MF,  we  have 

-  —  x  =  OI)—  OM  =  MD  =  PN, 

e 

-  +  x  =  D'0-{-  OM  =  D'M  =  N'P. 


Fig.  89 

The  ellipse  has  therefore  two  directrices  at  the  distances  ±  - 

e 

from  the  center.  When  the  ellipse  is  a  circle,  e  =  0  and  the 
directrices  are  at  infinity. 

In  a  similar  manner  we  may  show  that  the  hyperbola  is  a 
special  case  of  a  conic  where  e  >  1. 

In  §  114,  Ex.  3,  we  shall  prove  that  the  conic  is  always  either 
an  ellipse,  a  parabola,  or  an  hyperbola. 

82.  The  witch.  Let  OBA  (fig.  90)  be  a  circle,  OA  a  diameter, 
and  LK  the  tangent  to  the  circle  at  A.  From  0  draw  any  line 
intersecting  the  circle  at  B  and  LK  at  C.  From  B  draw  a  line 
parallel  to  LK  and  from  C  a  line  perpendicular  to  LK,  and  call 
the  intersection  of  these  two  Hnes  P.  The  locus  of  P  is  a  curve 
called  the  witch. 


150     CERTAIN  CURVES  AND  THEIR  EQUATIONS 

To  obtain  its  equation  we  wiU  take  the  origin  at  0  and  the 
line  OA  as  the  axis  of  y.  We  will  call  the  length  of  the  diameter 
of  the  circle  2  a.  Then  by  continuing  CP  until  it  meets  OX  at 
M,  and  calling  {x,  y)  the  coordinates  of  P,  we  have 


OM=x,         MP  =  y, 
MP      OB 


0A=MC=2a. 
OBOC 


In  the  triangle  OMC,  — —  =  — -  = 

°  MC      OC         OC 

If  AB  is  drawn,  OB  A  is  a  right  angle  and  consequently 
0B0C  =  0A\     also  OC'  =  OM^'+MC^ 


(1) 


Fig.  90 


Therefore 
that  is, 
and  finally. 


JifP  ^         OA 

^C~  OM^  +  MC^ 

y  __     4  a'^     . 
2a 


y 


x'+4.a' 


(2) 

(3) 
(4) 


Conversely,  if  equation  (4)  is  satisfied  by  any  point,  we  can 
deduce  equations  (3),  (2),  and  (1)  in  order,  and  hence  show  that 
the  point  is  on  the  witch. 

Solving  (4)  for  x,  we  have 


.  ^        \2a-y 


THE  CISSOID 


151 


This  shows  (1)  that  the  curve  is  symmetrical  with  respect  to  OY, 
(2)  that  y  cannot  be  negative  nor  greater  than  2  a,  and  (3)  that 
?/  =  0  is  an  asymptote. 

83.  The  cissoid.  Let  ODA  (fig.  91)  be  a  circle  with  the  diam- 
eter OA,  and  let  LK  be  the  tangent  to  the  circle  at  A.  Through 
0  draw  any  Ime  intersecting  the  circle  in  D  and  LK  in  E.  On 
OE  lay  off  a  distance  OP  equal  to  DE.  Then  the  locus  of  P  is 
a  curve  called  the  cissoid. 

To  find  its  equation,  we  will 
take  0  as  the  origin  of  coordinates 
and  OA  as  the  axis  of  x,  and  will 
call  the  diameter  of  the  circle  2  a. 
Draw  MP  perpendicular  to  OA. 
Then  if  ^  and  Z>  are  connected,  a 
triangle  ADE  is  formed  similar 
to  OMP ;  whence 

OP  _AE 
MP~  DE 

By    hypothesis    DE  =  OP. 
Therefore 

W^MPAE.  (2) 

Also,  in    the   similar   triangles 
OAE  and  0PM, 

AE  _  MP . 
0A~~  OM' 
whence,  from  (2), 

^2  _  OA  .  MP' 


or      X  -\-  y  = 


whence     y^  = 


OM 

2af, 


9.n.— 


This  equation  is  satisfied  by 
the  coordinates  of  any  point 
upon  the  cissoid. 


Fig.  $)1 


152      CERTAIN  CURVES  AND  THEIR  EQUATIONS 


Conversely,  if  we  assume  equation  (6),  we  may  deduce  (5)  and 
(4),  aud  then  by  aid  of  (1)  and  (3)  we  have  OP  =  DE. 

Therefore  (6)  is  the  equation  of  the  cissoid.    It  may  be  written 


y  =  ±x 


1       X 
\2a-x 


From  this  it  appears 
(1)  that  the  curve  is 
symmetrical  with  re- 
spect to  OX,  (2)  that 
no  value  of  x  can  be 
greater  than  2  a  or 
less  than  0,  and  (3)  that 
the  line  x  =  2a  is  an 
asymptote. 

84.  The  strophoid. 
Let  ZA' and  i?^' (fig.  92) 
be  two  straight  lines 
intersecting  at  right 
A  angles  at  0,  and  let  A 
be  a  fixed  point  on  LK. 
Through  A  draw  any 
straight  line  intersecting 
RS  in.  D,  and  lay  off  on 
AD  in  either  direction  a 
distance  DP  equal  to  OD. 
The  locus  of  P  is  a  curve 
called  the  strophoid. 

To  find  its  equation, 
take  LK  as  the  axis  of 
X  and  BS  as  the  axis  of 
y,  and  call  the  coordi- 
nates of  A  (a,  0).  By 
the  definition  the  point 
P  may  fall  in  any  one 
of  the  four  quadrants. 


THE  STROPHOID  153 

If  we  take  the  positive  direction  on  AD  as  measured  from  A 

towards  D,  we  have 

OD=PI> 

when  F  is  in  the  first  quadrant, 

OD  =  -PD 

when  F  is  in  the  second  quadrant, 

-OD  =  -FD 
when  F  is  in  the  third  quadrant,  and 

-OD  =  FD 

when  F  is  in  the  fourth  quadrant. 

These   four  equations   are  equivalent   to  the  single  equation 

od'=fi>\  (1) 

From  the  similar  triangles  OAD  and  APM, 
OD      MF  y 


AD      AF      V(ic- 

■cif+f 

FD      MO      OM 
AD       AO       OA 

X 

a 

f 

{x-af+y^      a" 

Hence  ^^_af+f-d^  ^^ 

is  an  equation  satisfied  by  any  point  on  the  curve.    Conversely, 
if  (2)  is  given,  (1)  may  be  deduced.    Therefore  (2)  is  the  equation 
of  the  strophoid. 
It  may  be  written 


=  ±Xy^ 


\a  —  x 
^  ^a-\-  X 


This  shows  (1)  that  the  curve  is  symmetrical  with  respect  to 
OX,  (2)  that  no  value  of  x  can  be  less  than  -  a  nor  greater  than 
+  a,  and  (3)  that  «  -  -  «  is  an  asymptote. 


154      CEKTAIN  CURVES  AND  THEIR  EQUATIONS 


85.  Examples.  The  use  of  the  equation  of  a  curve  in  solving 
problems  connected  with  the  curve  will  be  constantly  illustrated 
throughout  the  book.  The  following  examples  depend  upon  prin- 
ciples already  given. 

Ex.  1.  Prove  that  in  the  ellipse  the  squares  of  the  ordinates  of  any  two 
points  are  to  each  other  as  the  products  of  the  segments  of  the  major  axis  made 

by  the  feet  of  these  ordinates. 
We  are  to  prove  that  (fig.  93) 

A'Mi  ■  M2A ' 

Let  the  coordinates  of  Pi  be 
X  (p^ii  Vi)  ^-nd  let  those  of  P^  be 
(X2,  2/2)-    Then 


(g  +  xi){a  -  xi) 
Fig.  93  y\      a'^  -  x^      («  +  ^^H'^  -  ^) 

But  yi  =  MiPi,  a  +  Xi  =  A'O  +  OMi  =  A'Mi,  a-Xi=OA  -  OMi  =  MiA, 
y-i  =  M2P2,  a  +  X2  =  A'Mz,  a  —  X2  =  M2A.    Hence  the  proposition  is  proved. 

Ex.  2.  If  MiPi  is  the  ordinate  of  a  point  Pi  of  the  parabola,  y2  _  4  px^  and 
a  straight  line  di-av?n  through  the  middle  point  of  ilfiPi  parallel  to  the  axis  of  x 
cuts  the  curve  at  Q;  prove  that  the  intercept  of  the  line  MiQ  on  the  axis  of  y 
equals  §  MiPi. 

Let  the   coordinates  of    Pi  (fig.  94)   be 

y? 

(Xi,  yi).    Then  xi  =  —  from  the  equation  of 


the  parabola. 


4p 


By  construction,  the  ordinate  of  Q  is 


Vi 


Since  Q  is  on  the  parabola  its  abscissa  is 

found  by  placing  y  =  ~  in  y'^  =  4px.    The 
/    2  \ 

coordinates  of  Q  are  then  ( — -  ,  —)•    The 

\16p     2/ 
cobrdinates  of  M^  are  (xi,  0),  which  are  the 

same  as  (  — »  Ol.     Hence  the  equation  of 
MQ  is,  by  §  29, 


Fig.  94 


8px  +  3yiy-2y^  =  0. 
The  intercept  of  this  line  on  OF  is  §  yi  =  2  ilfiPi,  which  was  to  be  proved. 


PROBLEMS  155 

PROBLEMS 

1.  Find  the  equation  of  the  circle  having  the  center  (2,  —4)  and  the  radius  3. 

2.  Find  the  equation  of  the  circle  having  the  center  (— §?  ^)  and  the 
radius  6. 

3.  Find  the  equations  of  the  circles  having  the  line  joining  (2,  3)  and  ( —  3, 1) 
as  a  radius. 

4.  Find  the  equation  of  the  circle  having  the  line  joining  (a,  —  h)  and 
( —  a,  6)  as  a  diameter. 

5.  Find  the  equations  of  the  circles  of  radius  a  which  are  tangent  to  the 
axis  of  y  at  the  origin. 

6.  Find  tiie  equations  of  the  circles  of  radius  a  which  are  tangent  to  both 
coordinate  axes. 

7.  Find  the  equation  of  the  circle  having  as  a  diameter  that  part  of  the  line 
2x  —  Sy  +  6  —  0  which  is  included  between  the  coordinate  axes. 

8.  Find  the  center  and  the  radius  of  the  circle  x^  +  ?/2  +  4  a;  — 10  y  —  36  =  0. 

9.  Find  the  center  and  the  radius  of  the  circle  x^+y^-\-4x-Qy-\-l  =  0. 

10.  Find  the  center  and  the  radius  of  the  circle  3x^  +  3y^  —  9x  +  6y  —  2  =  0. 

1 1.  Find  the  center  and  the  radius  of  the  circle  5x^+  5y^  + 2x  —  4ty  +  1  =  0. 

12.  Prove  that  two  circles  are  concentric  if  their  equations  differ  only  in  the 
absolute  term. 

13.  Show  that  the  circles  x^  +  y^  +  2Gx  +  2  Fy  +  C  =  0  and  x^  +  y^  +  2  G'x 
+  2  Yy  +  C"  =  0  are  tangent  to  each  other  if 


V((?-G')2  +  (F-i?")2  =  Vg2  +  F2  _  c-  ±  VG'2  +  2f"2  _  c". 

14.  Find  the  equation  of  the  circle  which  passes  through  the  points  (0,  3), 
(3,  0),  (0,  0). 

15.  Find  the  equation  of  the  circle  circumscribing  the  triangle  with  the 
vertices  (0,  2),  (-  1,  0),  (0,  -  2). 

16.  Find  the  equation  of  the  circle  circumscribed  about  the  triangle  the  sides 
of  which  are  x  +  y  —  2  =  0,  9x  +  oj/  —  2  =  0,  2/  +  2x  —  1  =  0. 

17.  Find  the  equation  of  the  circle  passing  through  the  point  (—  2,  4)  and 
concentric  with  the  circle  x2  +  ?/2— 5x  +  4?/  —  1  =  0. 

18.  A  circle  which  is  tangent  to  both  coordinate  axes  passes  through  (4,  2). 
Find  its  equation. 

19.  The  center  of  a  circle  which  is  tangent  to  the  axes  of  x  and  y  is  on  the 
line  2x  —  3y  +  6  =  0.    What  is  its  equation  ? 

20.  A  circle  of  radius  5  passes  through  the  points  (2,  —  1)  and  (3,  —  2).' 
What  is  its  equation  ? 

21.  The  center  of  a  circle  which  passes  through  the  points  (1,  —  2)  and 
(-  2,  2)  is  on  the  line  8x-42/  +  9  =  0.    What  is  its  equation  ? 


156      CERTAIN  CURVES  AND  THEIR  EQUATIONS 

22.  A  circle  which  is  tangent  to  OX  passes  through  (-3,  2)  and  (4,  9). 
What  is  its  equation  ? 

23.  The  center  of  a  circle  which  is  tangent  to  the  two  parallel  lines  x  —  3  =  0 
and  a;  -  7  =  0  is  on  the  line  y  =  2  x  +  4.    What  is  its  equation  ? 

24.  The  center  of  a  circle  is  on  the  line  2  x  +  y  =  0.  The  circle  passes 
through  the  point  (4,  2)  and  is  tangent  to  the  line  4x-3?/-15  =  0.  What  is 
its  equation  ? 

25.  Find  the  equation  of  the  circle  circumscribing  the  isosceles  triangle  of 
which  the  altitude  is  4  and  the  base  is  the  line  joining  the  points  (—  3,  0)  and 
(3,  0). 

26.  Find  the  equation  of  the  ellipse  the  foci  of  which  are  (±3,  0)  and 
the  major  axis  of  which  is  8. 

27.  Find  the  equation  of  the  ellipse  the  foci  of  which  are  (0,  ±  2)  and  the 
major  axis  of  which  is  6. 

28.  Find  the  equation  of  an  ellipse  when  the  vertices  are  (±6,  0)  and  one 
focus  is  (4,  0). 

29.  Determine  the  semiaxes  o  and  h  in  the  ellipse  —  H =  1,  so  that  it  will 

pass  through  (1,  4)  and  (2,  —  3). 

30.  If  the  vertices  of  an  ellipse  are  (±  5,  0)  and  its  foci  are  (±3,  0),  find 
its  equation. 

31.  The  center  of  an  ellipse  is  at  the  origin  and  its  major  axis  lies  along  OX. 
If  its  major  axis  is  8  and  its  eccentricity  is  ^,  find  its  equation. 

32.  Find  the  equation  of  an  ellipse  when  its  center  is  at  the  origin,  one  focus 
at  the  point  (—  3,  0),  and  the  minor  axis  equal  to  8. 

33.  Find  the  equation  of  an  ellipse  the  eccentricity  of  which  is  ^  and  the 
foci  of  which  are  (0,  ±  6). 

34.  Given  the  ellipse  9x2  +  16y2  =  144.  Find  its  semiaxes,  eccentricity,  and 
foci. 

35.  Find  the  eccentricity  and  the  equation  of  an  ellipse,  if  the  foci  lie  half- 
way between  the  center  and  the  vertices,  the  major  axis  lying  on  OX. 

36.  Find  the  equation  of  an  ellipse  the  eccentricity  of  which  is  f  and  the 
ordinate  at  the  focus  is  5,  the  center  being  at  the  origin  and  the  major  axis 
lying  along  OX. 

37.  Find  the  equation  and  the  eccentricity  of  the  ellipse  if  the  ordinate 
at  the  focus  is  one  fourth  the  minor  axis. 

38.  Find  the  eccentricity  of  an  ellipse  if  the  line  connecting  the  positive  ends 
of  the  axes  is  parallel  to  the  line  joining  the  center  to  the  upper  end  of  the 
onlinate  at  the  left-hand  focus. 

39.  Find  the  equation  of  an  ellipse  when  the  foci  are  (±2,0)  and  the 
directrices  are  x  =  ±  5. 

40.  Given  the  ellipse  2x^-\-Zy^  =  \.   Find  its  semiaxes,  foci,  and  directrices. 


PKOBLEMS  157 

41.  Find  the  equation  of  an  hyperbola  if  the  foci  are  (±  3,  0)  and  the  trans- 
verse axis  is  4. 

42.  Find  the  equation  of  an  hyperbola  if  the  foci  are  (0,  ±  4)  and  the  trans- 
verse axis  is  4. 

43.  An  hyperbola  has  its  center  at  the  origin  and  its  transverse  axis  along 
OX.    If  its  eccentricity  is  ^  and  its  transverse  axis  is  5,  find  its  equation. 

44.  Find  the  equation  of  an  hyperbola  when  the  vertices  are  (±  4,  0)  and 
the  eccentricity  is  I. 

45.  Show  that  the  eccentricity  of  an  equilateral  hyperbola  is  equal  to  the 
ratio  of  a  diagonal  of  a  square  to  its  side. 

46.  Find  the  equation  of  an  hyperbola  the  vertices  of  which  are  halfway 
between  the  center  and  the  foci,  the  transverse  axis  lying  on  OX. 

47.  Find  the  equation  of  the  hyperbola  with  eccentricity  3  which  passes 
through  the  point  (2,  4),  its  axes  lying  on  OX  and  OY. 

48.  Find  the  equation  of  an  equilateral  hyperbola  which  passes  through 
(5,  -  2). 

49.  Find  the  equation  of  the  hyperbola  which  has  the  points  (0,  ±  f  V2)  for 
foci  and  passes  through  the  point  (2,   —  1). 

50.  The  sum  of  the  semiaxes  of  an  hyperbola  is  17  and  its  eccentricity  is 
\^.    Find  its  equation,  if  its  axes  lie  on  OX  and  OY. 

51.  Find  the  equation  of  the  hyperbola  which  has  the  asymptotes  y  =  ±  ^x 
and  passes  through  the  point  (1,  1). 

52.  Express  the  angle  between  the  asymptotes  in  terms  of  the  eccentricity 
of  the  hyperbola. 

53.  If  the  vertex  of  an  hyperbola  lies  two  thirds  of  the  distance  from  the 
center  to  the  focus,  find  the  slopes  of  the  asymptotes. 

54.  Given  the  hyperbola  4x^  —  25]/^=  100.    Find  its  eccentricity,  foci,  and 
asymptotes. 

55.  Find  the  equation  of  the  hyperbola  which  has  the  lines  y  =  ±  §  x  for 
its  asymptotes  and  the  points  ( ±  4,  0)  for  its  foci. 

56.  Show  that 1 =  1,  where  k  is  an  arbitrary  quantity, 

a^  —  k^     62  _  j(2 

x2      v^ 
represents  an  ellipse  confocal  to  — I-  —  =  l,when  ^2  <;  52  j  and  represents  an 

a;2      y2       a^      t>'^ 
hyperbola  confocal  to  —  +  ^=  1,  when  k'^  >62  but<  a"^,  a^  being  considered 

greater  than  &2.  «^      ^^ 

57.  Find  the  equation  of  an  hyperbola  when  the  foci  are  (±  7,  0)  and  the 
directrices  are  x  =  ±  4. 

X2         ?/2 

58.  Given  the  hyperbola —  =  1.    Find  its  eccentricity,  foci,  directrices, 

and  asymptotes. 


158      CERTAIN  CURVES  AND  THEIR  EQUATIONS 

59.  A  perpendicular  is  drawn  from  a  focus  of  an  hyperbola  to  an  asymptote. 
Show  that  its  foot  is  at  distances  a  and  b  from  the  center  and  the  focus 
respectively. 

60.  Show  that  in  an  equilateral  hyperbola  the  distance  of  a  point  from  the 
center  is  a  mean  proportional  "between  its  focal  distances. 

61.  Determine  p  so  that  the  parabola  y'^  =  ipx  shall  pass  through  the  point 
(2,  -  3). 

62.  An  arch  in  the  form  of  a  parabolic  curve  is  29  ft,  across  the  bottom  and 
the  highest  point  is  8  ft.  above  the  horizontal.  What  is  the  length  of  a  beam 
placed  horizontally  across  the  arch  4  ft.  from  the  top  ? 

63.  The  cable  of  a  suspension  bridge  hangs  in  the  form  of  a  parabola.  The 
roadway,  which  is  horizonUl  and  240  ft.  long,  is  supported  by  vertical  wires 
attached  to  the  cable,  the  longest  being  80  ft.  and  the  shortest  being  30  ft. 
Find  the  length  of  a  supporting  wire  attached  to  the  roadway  50  ft.  from  the 
middle. 

64.  Find  the  equation  of  a  circle  through  the  vertex  and  the  ends  of  the 
double  ordinate  through  the  focus  of  the  parabola  y^  =  A  px. 

65.  Find  the  equation  of  the  circle  through  the  vertex,  the  focus,  and  the 
upper  end  of  the  ordinate  at  the  focus,  of  the  parabola  y^  +  12x  =  0. 

66.  Find  the  equation  of  the  locus  of  a  point  the  distances  of  which  from 
(8,  -  2)  and  (-4,  1)  are  equal. 

67.  Find  the  equations  of  the  locus  of  a  point  the  distance  of  which  from  the 
axis  of  X  equals  five  times  the  distance  from  the  axis  of  y. 

68.  Find  the  equation  of  the  locus  of  a  point  the  distance  of  which  from  the 
axis  of  X  is  one  third  its  disUnce  from  (0,  3). 

69.  Find  the  equation  of  the  locus  of  a  point  the  distance  of  which  from 
the  line  x  =  3  is  equal  to  its  distance  from  (4,  -  2). 

70.  What  is  the  locus  of  a  point  the  distance  of  which  from  the  line 
3x  +  4y  —  6  =  0  is  twice  its  distance  from  (2,  1)  ? 

71.  A  point  moves  so  that  its  distance  from  the  axis  of  y  equals  its  distance 
from  the  point  (5,  0).    Find  the  equation  of  its  locus. 

72.  A  point  moves  so  that  the  square  of  its  distance  from  the  point  (0,  2) 
equals  the  cube  of  its  distance  from  the  axis  of  y.    Find  its  locus. 

73.  Find  the  locus  of  the  points  at  a  constant  distance  6  from  the  line 
4x  +  Sy  -6  =  0. 

74.  Find  the  locus  of  points  equally  distant  from  the  lines  2x  +  3w-6  =  0 
and  Sx  —  2y  +  1  =  0. 

75.  Show  that  the  locus  of  a  point  which  moves  so  that  the  sum  of  its  dis- 
tances from  two  fixed  straight  lines  is  constant  is  a  straight  line. 

76.  Find  the  equations  of  the  locus  of  a  point  equally  distant  from  two  fixed 
straight  lines. 


PROBLEMS  159 

77.  A  point  moves  so  that  its  distances  from  two  fixed  points  are  in  a  con- 
stant ratio  k.    Sliow  tliat  tlie  locus  is  a  circle  except  when  k  —  \. 

78.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from  the 
sides  of  an  equilateral  triangle  is  constant.  Show  that  the  locus  is  a  circle  and 
find  its  center. 

79.  A  point  moves  so  that  the  square  of  its  distance  from  the  base  of  an 
isosceles  triangle  is  equal  to  the  product  of  its  distances  from  the  other  two 
sides.  Show  that  the  locus  is  a  circle  and  an  hyperbola  which  pass  through  the 
vertices  of  the  two  base  angles. 

80.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from  the 
four  sides  of  a  square  is  constant.    Find  its  locus. 

81.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from  any 
number  of  fixed  points  is  constant.    Find  its  locus. 

82v  Find  the  locus  of  a  point  the  square  of  the  distance  of  which  from  a 
fixed  point  is  proportional  to  its  distance  from  a  fixed  straight  line. 

83.  Find  the  locus  of  a  point  such  that  the  lengths  of  the  tangents  from  it 
to  two  concentric  circles  are  inversely  as  the  radii  of  the  circles. 

84.  A  point  moves  so  that  the  length  of  the  tangent  from  it  to  a  fixed  circle 
is  equal  to  its  distance  from  a  fixed  point.    Find  its  locus. 

85.  Find  the  equation  of  the  locus  of  a  point  the  tangents  from  wliich  to 
two  fixed  circles  are  of  equal  length. 

86.  Straight  lines  are  drawn  through  the  points  (—  a,  0)  and  (a,  0)  so  that 

the  difference  of  the  angles  they  make  with  the  axis  of  x  is  tan- 1  - .    Find  the 
locus  of  their  point  of  intersection. 

87.  The  slope  of  a  straight  line  passing  through  (a,  0)  is  twice  the  slope  of 
a  straight  line  passing  through  (—  a,  0).  Find  the  locus  of  the  point  of  inter- 
section of  these  lines. 

88.  A  point  moves  so  that  the  product  of  the  slopes  of  the  straight  lines 
joining  it  to  A  {—a,  0)  and  B  («,  0)  is  constant.  Prove  that  the  locus  is  an 
ellipse  or  an  hyperbola. 

89.  If,  in  the  triangle  ABC,  taw  A  tan^B  =  2  and  AB  is  fixed,  show  that 
the  locus  of  C  is  a  parabola  with  its  vertex  at  A  and  focus  at  B. 

90.  Given  the  base  2  6  of  a  triangle  and  the  sum  s  of  the  tangents  of  the 
angles  at  the  base.    Find  the  locus  of  the  vertex. 

91.  Find  the  locus  of  the  center  of  a  circle  which  is  tangent  to  a  fixed  circle 
and  a  fixed  straight  line. 

92.  Prove  that  the  locus  of  the  center  of  a  circle  which  passes  through  a 
fixed  point  and  is  tangent  to  a  fixed  straight  line  is  a  parabola. 

93.  A  point  moves  so  that  its  shortest  distance  from  a  fixed  circle  is  equal  to 
it8  distance  from  a  fixed  diameter  of  that  circle.    Find  its  locus. 


160      CERTAIN  CURVES  AXD  THEIR  EQUATIONS 

94.  0  is  a  fixed  point  and  AB  is  a  fixed  straight  line.  A  straight  line 
is  drawn  from  0  meeting  AB  at  Q,  and  in  OQ  a  point  P  is  taken  so  that 
OPOQ  =  k^.    Find  the  locus  of  P. 

95.  If  a  straight  line  is  drawn  from  the  origin  to  any  point  Q  of  the  line 
y  =  a,  and  if  a  point  P  is  taken  on  this  line  such  that  its  ordinate  is  equal  to 
the  abscissa  of  Q,  find  the  locus  of  P. 

96.  A  OB  and  COD  are  two  straight  lines  which  bisect  each  other  at  right 
angles.    Find  the  locus  of  a  point  P  such  that  PA  •  PB  =  PC  •  PD. 

97.  AB  and  CD  are  perpendicular  diameters  of  a  circle  and  M  is  any  point 
on  the  circle.  Through  Jf,  AM  and  BM  are  drawn.  AM  intersects  CD  in  N, 
and  from  J\r  a  line  is  drawn  parallel  to  AB  meeting  BM  in  P.    Find  the  locus  of  P. 

98.  Given  a  fixed  line  AB  and  a  fixed  point  Q.  From  any  point  R  in  AB  a 
perpendicular  to  AB  is  drawn  equal  in  length  to  RQ.  Find  the  locus  of  the  end 
of  this  perpendicular. 

99.  Let  OA  be  the  diameter  of  a  fixed  circle.  From  J5,  any  point  on  the 
circle,  draw  a  line  perpendicular  to  OA  and  meeting  it  in  D.  Prolong  the  line 
Z>B  to  P,  so  that  OD:DB=OA:  DP.    Find  the  locus  of  P. 

100.  Two  straight  lines  are  drawn  through  the  vertex  of  a  parabola  at  right 
angles  to  each  other  and  meeting  the  curve  at  P  and  Q.  Show  that  the  line  PQ 
cuts  the  axis  of  the  parabola  in  a  fixed  point. 

101.  In  the  parabola  y^  =  ^px  an  equilateral  triangle  is  so  inscribed  that 
one  vertex  is  at  the  origin.    What  is  the  length  of  one  of  its  sides  ? 

102.  Prove  that  in  the  ellipse  half  of  the  minor  axis  is  a  mean  proportional 
between  AF  and  FA'. 

103.  Prove  that  in  the  ellipse  or  the  hyperbola  the  ordinate  at  the  focus 
is  an  harmonic  mean  between  AF  and  AF'. 

104.  If  from  any  point  P  of  an  hyperbola  PK  is  drawn  parallel  to  the 
transverse  axis,  cutting  the  asymptotes  in  Q  and  R,  then  PQ-  PR-  a^.  If  PK 
is  drawn  parallel  to  the  conjugate  axis,  then  PQ  ■  PR  =  -  b-. 

105.  Show  that  the  focal  disUnce  of  any  point  on  the  hyperbola  is  equal  to 
the  length  of  the  straight  line  drawn  through  the  point  parallel  to  an  asymptote 
to  meet  the  corresponding  directrix. 

106.  Prove  that  the  product  of  the  distances  of  any  point  of  the  hyperbola 
from  the  asymptotes  is  constant. 

107.  Prove  that  in  the  hyperbola  the  squares  of  the  ordinates  of  any  two 
points  are  to  each  other  as  the  products  of  the  segments  of  the  transverse  axis 
made  by  the  feet  of  these  ordinates. 

108.  Lines  are  drawn  through  a  point  of  an  ellipse  from  the  two  ends  of 
the  minor  axis.    Show  that  the  product  of  their  intercepts  on  OX  is  constant. 

109.  Pi  is  any  point  of  the  parabola  y^  =  4px,  and  P^Q,  which  is  perpen- 
dicular to  OPi,  intersects  the  axis  of  the  parabola  in  Q.  Prove  that  the  pro- 
jection of  PiQ  on  the  axis  of  the  parabola  is  always  4 p. 


CHAPTEE  VIII 
INTERSECTION    OF  CURVES 

86.  General  principle.    If  f„J(x,  y)  is  an  expression  involving 

"^"'^-  /..(»=.  y)  =  o  (1) 

is  the  equation  of  a  curve  containing  all  points  the  coordinates 
of  which  satisfy  (1),  and  containing  no  other  points.  Similarly  if 
/„(ie,  y)  is  any  second  expression  in  x  and  y, 

/„(^,^)  =  0  (2) 

is  the  equation  of  a  second  curve.  It  follows  that  if  we  consider 
these  two  equations,  any  point  common  to  the  two  corresponding 
curves  will  have  coordinates  satisfying  both  (1)  and  (2) ;  and  that, 
conversely,  any  values  of  x  and  y  which  satisfy  both  (1)  and  (2) 
are  coordinates  of  a  point  common  to  the  two  curves.  Hence, 
to  find  the  joints  of  intersection  of  two  curves,  solve  their  equa- 
tions simultaneously. 

We  have  already  discussed  in  §  30  the  simplest  case  of  this 
problem,  i.e.  the  intersection  of  two  straight  lines.  We  shall  now 
discuss  some  more  complex  cases. 

87.  /,(x,  t/)  =  Oand/,(;c,  (/)  =  0.    Let 

fM,y)=^  (1) 

be  a  linear  equation,  and     f^  {x,  y)=0  (2) 

be  a  quadratic  equation.  Since  a  linear  equation  always  represents 
a  straight  line,  this  problem  is  to  find  the  points  of  intersection  of  a 
straight  Ime  and  a  curve.  Solving  (1)  for  either  x  or  y,  and  substitut- 
ing the  result  in  (2),  we  obtain  in  general  a  quadratic  equation,  as, 
for  example,  aa^  +  hx  +  c=:0, 

if  (1)  has  been  solved  for  y.  We  shall  call  this  equation  the 
resultant  equation  {§  9).    If  the  roots  of  this  equation  are  denoted 

161 


162 


INTERSECTION  OF  CURVES 


by  x^  and  x^,  x^  and  x^  are  the  abscissas  of  the  required  points  of 
intersection.  The  corresponding  ordinates  are  found  by  substi- 
tuting x^  and  x^  in  succession  in  (1). 

But  according  to  §  37  there  are  three  cases  to  be  considered  in 
the  solution  of  the  resultant  equation.  (1)  The  roots  x^  and  x^ 
may  be  real  and  unequal,  in  which  case  there  are  two  points  of 
intersection.  (2)  The  roots  x^  and  x^  may  be  real  and  equal,  in 
which  case  the  corresponding  ordinates  are  equal  and  the  two 
points  coincide.  As  in  §  37,  we  may  regard  this  case  as  a  limit- 
ing case  when  the  position  of  the  curves  is  changed  so  as  to  make 
x^  and  x^  approach  each  other,  i.e.  so  as  to  make  the  points  of  inter- 
section of  the  straight  line  and  the  curve  approach  each  other  along 
•the  curve.  Accordingly,  the  straight  line  represented  by  equation  (1) 
is  tangent  to  the  curve  represented  by  equation  (2).  (3)  Finally,  the 
roots  x^  and  x^  may  be  imaginary,  in  which  case  no  real  points  of 
intersection  can  be  found,  and  the  curves  do  not  intersect. 


Ex. 


and 


1.    Find  the  points  of  intersection  of 

3x-2y-4  =  0 

r 


(1) 
(2) 


Fig. 96 

Solving  (1)  for  y  and  substituting  tlie  result  in  (2),  we  have  x^  -  6x  +  8  =  0, 
the  roots  of  which  are  2  and  4.    Substituting  these  values  of  x  in  (1),  we  find  the 


STRAIGHT  LIKE  AND  COKIC 


163 


corresponding  values  of  y  to  be  1  and  4.    Therefore  the  points  of  intersection 
are  (2,  1)  and  (4,  4)  (fig.  95). 

Ex.  2.    Find    the    points    of 
intersection  of 


and 


6x-4y  -9  =  0 
a;2  -  4  2/  =  0. 


(1) 
(2) 


Solving  (1)  for  y  and  substi- 
tuting the  result  in  (2),  we  have 
x2-6x  +  9  =  0.  The  roots  of 
this  equation  are  equal,  each 
being  3.  Hence  the  straight  line 
is  tangent  to  the  curve.  Substi- 
tuting 3  for  X  in  (1),  we  find 
2/  =  I  ;  hence  the  point  of  tan- 
gency  is  (3,  f )  (fig.  96). 

Ex.  3.  Find  the  points  of 
intersection  of 

3x-2y-5  =  0  (1) 

and  x2  -  4  y  =  0.  (2) 

Proceeding  as  in  the  two  previous  examples,  we  obtain  x^  —  6  x  +  10  =  0, 
the  roots  of  which  are  3  ±  V  — 1.    Hence  the  straight  line  does  not  intersect  the 

curve  (fig.  97).    The  correspond- 
ing values  of  y  are  2  ±  |  V—  1. 

It  is  to  be  noted  that  the 
straight  lines  of  these  three 
examples  all  have  the  same 
direction,  differing  only  in  the 
intercept  on  the  axis  of  y. 


88.  The  work  of  the  last 
article  suggests  a  method 
'^  of  finding  the  tangent  to 
any  curve  represented  by 
an  equation  of  the  second 
degree,  the  slope  of  the 
tangent  being  given.  For 
if  m  of  the  required  tan- 
gent is  known,  its  equation  may  be  written  y  =  mx  +  b,  where 
h  is  not  known.  According  to  the  definition  of  a  tangent,  how- 
ever, b  must  have  such  value  that  the  points  of  intersection  of 


Fig.  97 


164 


INTERSECTION  OF  CURVES 


straight  line  and  curve  shall  be  coincident.    This  condition  enables 
us  to  determine  b,  as  shown  in  the  following  examples. 


Ex.  1.  Find  the  equation  of  tlie 
tangent  to  the  parabola  3  x^  +  2  7/ =0, 
the  slope  of  the  tangent  being  2. 

Since  the  slope  of  the  tangent 
is  2,  its  equation  may  be  written 
y  =  2x  +  b.  Substituting  this  value 
of  y  in  the  equation  of  the  parabola, 
•we  have  the  equation 

3a;2  +  4x  +  26  =  0. 

Since  the  line  is  to  intersect  the 
curve  in  two  coincident  points,  this 
equation  must  have  equal  roots. 
The  condition  for  equal  roots,  by 
§  .37,  is  (4)2  -  4  (.3)  (2  b)  =  0,  whence 
we  find  6  =  |-. 


=  2x  +  |-,  or  6x-Sy  +  2  =  0  (fig. 


Fig.  98 
Therefore  the  required  tangent  is 

Ex.  2.  Find  the  equation 
of  the  tangent  to  the  ellipse 
x2  -I-  4  y2  =  4,  the  slope  of  the 
tangent  being  ^. 

The  equation  of  the  tangent 
is  y  =  ^z  +  b.  Substituting  this 
value  of  y  in  the  equation  of  the 
ellipse,  we  have 

x2  +  2  6a;  +  (2  62  -  2)  =  0. 

Fig.  99 
The  condition  that  this  equation 

shall  have  equal  roots  is  (2  6)2  -  4  (2  62  -  2)  =  0,  whence  6  =  ±  V2. 

In  this  case  there  are  two  tangents  having  the  required  slope  -^  (fig.  99),  the 

equations  of  which  are  respectively  j^  =  i  x  +  Vi  and  y  =  ^x-V2 

or  x-2y±2V2  =  0. 

By  this  same  method  the  following  formulas  for  a  tangent  with 
known  slope  m  may  be  derived : 

1.  The  tangent  to  the  parabola  y-  =  Apx  is 


y  =  mx  + 


EXCEPTIONAL  CASES 


165 


2.  The  tangents  to  the  ellipse  —  +  ^  =  1  are 


=  mx  ±  Va^m^  +  b^. 


X  IT 

3.  The  tangents  to  the  hyperbola  —  —  4  =  1  are 

a-     ■  l)- 


y 


=  mx  ±  ^ d^m^—  If. 


89.  It  was  stated  in  §  87  that  the  result  of  the  substitution  is 
in  general  a  quadratic  equation.  In  exceptional  cases,  however,  the 
resultant  equation  may  be  linear, 
as  in  the  first  of  the  following 
examples,  or  even  impossible,  as 
in  the  second  example. 

Ex.  1.    Consider 

2a; -5?/ -10  =  0  (1) 

and  4x2-252/2  =  100.        (2) 


Fig.  100 


Substituting  in  (2)  tlie  value  of  y 
from  (1),  we  have  tlie  equation  40  x  —  200  =  0,  wlience  x  =  5.     .-.  y  =  0,  and 
tlie  straight  line  and  the  curve  intersect  in  a  single  point  (o,  0)  (tig.  100). 


E.\.  2.    Consider 
and 


2x-5y  +  4  =  0 
4x2 -25?/2  +  i6x- 84  =  0. 


(1) 
(2) 


Fig.  101 

Substituting  in  (2)  the  value  of  y  from  (1),  we  have  —  100  =  0.  But  this 
equation  is  impossible.  Hence  the  given  equations  are  contradictory,  and  the 
straight  line  and  the  curve  do  not  intersect  (fig.  101). 


166  INTERSECTION  OF  CURVES 

These  exceptional  cases,  of  which  the  above  are  illustrative 
examples,  may  be  regarded  as  limiting  cases  as  follows : 

If  x^  and  x^  are  the  roots  of  the  resultant  equation  aaP  +  bx  +  c  =  0, 

2c 

X,  = 


_5  +  v^ 

-  4ac 

2a 

-&-V&^- 

-4ac 

-6-V&--4ac 
2c 


2a  —h  -f  Vft^— 4 


ac 


Now  as  a  ==  0,  the  resultant  equation  approaches  the  linear  equa- 
tion &«  +  c  =  0.    At  the  same  time  x.= and  a;„  =  cx) .    There- 

l  ^ 

fore,  if  a  is  made  to  approach  zero  by  changing  the  position  of 
either  the  straight  line  or  the  curve  in  the  plane,  the  case  in  which 
only  one  solution  of  the  linear  and  the  quadratic  equations  is 
found  is  the  limiting  case  of  intersection  of  the  straight  line  and 
the  curve  as  one  point  of  intersection  recedes  indefinitely  from 
the  origin. 

If  a  =  0  and  &  =  0,  both  x^  and  x^  increase  indefinitely.  Hence 
the  case  in  which  the  linear  and  the  quadratic  equations  are  con- 
tradictory is  the  limitmg  case  of  intersection,  as  both  points  of 
intersection  recede  indefinitely  from  the  origin. 

90-  /i(^,  1/)  =  0  and  /„(x,  y)  =  0.    Let 

/i(-^>  y)  =  0  (1) 

be  a  linear  equation,  and      fj^x,  y)  =  0  (2) 

be  an  equation  of  the  wth  degree  where  w  >  2.  The  degree  of  a 
curve  is  defined  as  equal  to  the  degree  of  its  equation.  Accord- 
ingly, this  problem  is  to  find  the  points  of  intersection  of  a  straight 
line  and  a  curve  of  the  w,th  degree  where  %->%  and  the  method 
is  the  same  as  that  of  §  87.  The  resultant  equation,  after  sub- 
stitution from  the  linear  equation,  is,  in  general,  of  the  wth  degree, 
and  its  solution  is  found  by  the  methods  of  Chaps.  IV  and  V. 
The  number  of  points  of  intersection  will  be  the  same  as  the 
number  of  real  roots  of  the  resultant  equation.    Hence  a  straight 


STRAIGHT  LINE  AND  CURVE  OF  A^th  DEGREE     167 


line  can  intersect  a  curve  of  the  nth  degree  in  n  points  at  most. 
If  the  resultant  equation  has  multiple  roots,  they  correspond,  in 
general,  to  points  of  tangency  of  the  straight  line  and  the  curve, 
as   in    §  88 ;    and   if    the    resultant 
equation  is  of  degree  less  than  n,  it 
can  be  shown  that  the  straight  line  is 
the  limiting  position  of  one  in  which 
one  or  more  points    of   intersection 
have  been  made  to  recede  indefinitely. 

Ex.  1.    Find  the  points  of  intersection  of 

y  =  2x  (1) 

and  y2  =  x(x-3)2.  (2) 

The  resultant  equation  is 

a;[(x-3)2-4x]  =  0, 
or  a;[x2-10x  + 9]  =  0. 

Its  roots  (§  39)  are  the  roots  of  x  =  0  and 
x2-10x  +  9  =  0,  which  are  0,    1,   and  9. 

The  corresponding 

values  of  y  are  found 

from  (1)  to  be  0,  2, 

and  18.  Therefore  the 

points  of  intersection 

are  (0,  0),  (1,  2),  and 

(9,  18)  (fig.  102). 


Ex.  2.  Find  the 
points  of  intersection 
of 

y  =  3x  +  2{l) 

and      y  =  x^.         (2) 

Fig.  102 
The  resultant  equa- 
tion isx^  — 3x  — 2  =  0. 

One  root  is  found  (§  49)  to  be  2,  and  the  depressed 
equation  isx2  +  2x  +  1  =  0.  Its  roots  are  equal,  both 
being  -  1.  The  corresponding  values  of  y,  found 
from  (1),  are  8  and  -  1.  Therefore  the  points  of 
intersection  are  (2,  8)  and  (-  1,  -  1),  the  latter  being 
a  point  of  tangeucy  (fig.  103). 


Fig.  103 


168 


INTERSECTION  OF  CURVES 


Ex.  3.    Find  the  points  of  intersection  of 

2x  +  2/-4  =  0  (1) 

and  y2  =  x(x2-12).  (2) 

The  resultant  equation  is  x^  -  4x2  +  4x  -  16  =  0,  or  (x  -  4)  {x^  -f  4)  =  0, 
the  roots  of  which  are  4  and  ±  2  -Z^.    The  corresponding  values  of  2/,  found 


Fig.  104 

from  (1),  are  —  4  and  4^4  V—  1.  The  only  real  solution  of  equations  (1)  and 
(2)  being  x  =  4  and  i/  =  —  4,  the  straight  line  and  the  curve  intersect  in  the  single 
point  (4,  -  4)  (fig.  104). 

91. /„(x,  i/)=Oand/„(;t,  i/)=0.    Let 

LM^y)  =  ^  (1) 

be  an  equation  of  the  wth  degree  and 

/„(•«,  2/)  =0  (2) 

be  an  equation  of  the  nth  degree,  where  m  and  n  are  both  greater 
than  unity.  The  method  is  the  same  as  in  the  preceding  cases,  i.e. 
the  elimination  of  either  x  or  y,  the  solution  of  the  resultant  equa- 
tion, and  the  determination  of  the  corresponding  values  of  the 
unknown  quantity  eliminated.    The  equation  resulting  from  the 


CURVES  OF  Mth  AND  Nth  DEGREE 


169 


elimination  is,  in  general,  of  degree  mn,  and  the  number  of  simul- 
taneous solutions  of  the  original  equations  is  mn.  If  all  these 
solutions  are  real,  the  corresponding  curves  intersect  at  m7i  points. 
If,  however,  any  of  these  solutions  are  imaginary,  or  are  alike,  if 
real,  the  corresponding  curves  will  intersect  at  a  number  of  points 
less  than  'tnn.  Hence,  two  curves  of  degrees  m  and  n  respectively 
can  intersect  at  mn  points  and  no  more. 

No  attempt  at  a  complete  discussion  will  be  made,  on  account 
of  the  unlimited  number  of  cases  which  are  possible.  We  shall 
merely  solve  a  few  illustrative  examples,  noting  any  interesting 
geometrical  facts  that  may  occur  in  the  course  of  the  sokition: 

Ex.  1.  Find  the  points  of  intersection 
of 

y2_2x  =  0  (1) 

and  x2  +  2/2_8  =  o.  (2) 

Subtracting  (1)  from  (2),  we  elimi- 
nate y,  thereby  obtaining  the  resultant 
equation  a;^  +  2  a;  —  8  =  0,  the  roots  of 
which  are  —  4  and  2.  Substituting  2 
and  —  4  in  either  (1)  or  (2),  we  find  the 
corresponding  values  of  ?/  to  be  ±  2  and 
±  2  V—  2.  The  real  solutions  of  the 
equations  are  accordingly  x  =  2,  y  =  ±2, 
and  the  corresponding  curves  intersect 
at  the  points  (2,  2)  and  (2,  -  2)  (fig.  105). 

From  the  figure  it  is  also  evident  that 
the  value  —  4  for  x  must  make  y  imaginary,  as  both  curves  lie  entirely  to  the 

right  of  the  line  x  =  —  4. 
Y  ° 

Ex.  2.  Find  the  points  of  inter- 
section of 

x2  -  3  2/  =  0  (1) 

and  2/2_3x  =  0.  (2) 

Substituting  in  (2)  the  value  of  y 
from  (1),  we  have  x*  — 27x  =  0. 
This  equation  may  be  written 

x(x-3)(x2  +  3x  +  0)  =  0, 

the   roots   of   which   are   0,  3,  and 

_  3  4-  3  V  —  3 

— Substituting    these 

values  of  x  in  (1).  we  find  the  corre- 
Fio.  lOG  spending  values  of  y   to  be  0,   3, 


Fig.  105 


-X 


170 


INTEKSECTION  OF  CUEVES 


and 


-  3^8^^ 


Therefore  the  real  solutions  of  these  equations  are  a;  =  0, 


y  =  0,  and  x  =  3,  y  =  3.  If  we  had  substituted  the  values  of  x  in  (2),  we  should 
have  at  first  seemed  to  find  an  additional  real  solution,  y  —  —  S  when  x  =  3. 
But  —  3  for  2/  makes  x  imaginary  in  (1),  as  no  part  of  (1)  is  below  the  axis 
of  X.  Geometrically,  the  line  x  =  3  intersects  the  curves  (1)  and  (2)  in  a 
common  point  and  also  intersects  (2)  in  another  point.  Therefore  the  only 
real  solutions  of  these  equations  are  the  ones  noted  above,  and  the  corre- 
sponding curves  intersect  at  the  two  points  (0,  0)  and  (3,  3)  (fig.  106). 

We  8ee,  moreover,  that  any  results  found  must  be  tested  by  substitution  in  both 
of  the  original  equations. 

The  remaining  two  solutions  of    these  equations  found  by  letting  x  = 

_3±3Vr3 
are  imaginary. 


Ex.  3.    Find  the  points  of  intersection  of 

2x2  +  32/2  =  35 
and  xy  =  6. 


(1) 
(2) 


Since  these  equations  are  homogeneous  quadratic  equations  we  place 

y  =  mx  (3) 

and  substitute  for  y  in  both  (1)  and  (2).    The  results  are  2  x^  +  3  m'h?  =  35  and 


mx2 

=  6,  whence 

x2  = 

35 

(4) 

2  +  3m2 

and 

x2  = 

6 
m 

(6) 

35 

6 

5 

m 

(6) 

2  +  3  jft2 

from  which  we 

find  m  = 

i 

or  4. 

Fig.  107 


If  m  =  ^,  from  (5)  x  =  ±  2  ;  and 
from  (3)  the  corresponding  values 
of  y  are  ±  3. 

If  m  =  |,  in  like  manner  we  find 

X  =  ±  '^  Vg  and  y  —  ±^  V6. 

Therefore  the  ellipse  and  the  hyperbola  intersect  at  the  four  points  (2,  3), 
(-  2,  -  3),  (I  V6,  f  V6),  (-  I  V6,  -  I  V6)  (fig.  107). 

It  should  be  noted  that  (3)  is  the  equation  of  a  straight  line  through  the 
origin,  so  that  when  we  solve  (6)  for  m  we  determine  the  slopes  of  the  straight 
lines  passing  through  the  origin  and  intersecting  the  two  given  curves  at  their 
common  points. 


SYSTEMS  OF  CURVES 


171 


Ex.  4.  Find  the  points  of  inter- 
section of 

2  2/2  =  x-2  (1) 

and         x2  -  4  2/2  =  4.  (2) 

Eliminating  y,  we  have 


a;'^ 


2a;  =  0, 


Fig.  108 


the  roots  of  which  are  0  and  2. 

Wlien  X  =  0  we  find  from  either 

(1)  or  (2)  y  =  ±  V—  1,  and  when 

X  =  2  eitlier   (1)   or  (2)  reduces  to  y'^  =  0,  whence   y  =  0. 

two  curves  are  taijgent  at  the  point  (2,  0)  (fig.  108). 


Therefore  these 


Ex.  5.  Find  the  points  of  intersec- 


uuxiux                              x2  =  2y 

(1) 

and        x^  —  3  XT/  +  2/3  =  0. 

(2) 

Eliminating  j/,  we  have 

x6- 4x3  =  0, 

which  may  be  written  x^  (x^  —  4)  =  0. 

X  The  real  roots  of  this  equation  are  4^ 

and  0,  the  latter  being  a  triple  root, 

and  the   two   imaginary   roots   are 

4*(-l±V-3)     ^  ,. 

— i -•    Corresponding 

2 
values  of  y  are  found  to  be  2^,  0,  and 
2-^(- 1  T  ^^^^)-      Therefore    the 
Fig.  109  curves  intersect  at  the  two  points 

(4*,  2i)  and  (0,  0)  (fig.  109). 
At  the  point  (0,  0)  the  parabola  (1)  is  tangent  to  one  part  of  (2)  and  passes 
through  another  part  of  (2),  and  for  this  reason  the  point  is  to  be  regarded  as  a 
triple  point  of  intersection. 


^^'  Ifmi^t  y)~^  kfn{x,  i/)=0.  If  we  have  two  expressions 
f^{x,  y)  and  f,^{x,  y),  we  have  seen  in  §  86  that  we  can  form  the 
equations  of  two  curves  by  placing  each  of  them  separately  equal 
to  zero,  i.e. 

A(^,2/)  =  0,  (1) 

and  /„(^,2/)  =  0.  (2) 

Let  us  now  form  the  equation  of  a  tliird  curve  by  multiplying 
fjx,  y)  and  fjx,  y)  by  I  and  k  respectively,  where  I  and  k  are 


172 


INTERSECTION  OF  CURVES 


any  two  quantities  which  are  independent  of  both  x  and  y,  and 
placing  the  sum  of  the  products  equal  to  zero,  i.e. 


ifj<^>y)+¥n{^>y)  =  ^- 


(3) 


This  third  curve  has  the  following  two  important  properties : 

1.  It  passes  through  all  points  common  to  curves  (1)  and  (2). 
For  the  coordinates  of  any  such  points  make  /„,(a?,  y)  =  ^  and 
fj^x,  y)  =  0,  since  they  satisfy  (1)  and  (2).  Hence  they  will 
satisfy  (3),  ie.  be  coordinates  of  a  point  of  curve  (3). 

If  either  /  or  A;  is  placed  equal  to  zero,  (3)  reduces  to  either  (2) 
or  (1)  as  a  special  case. 

2.  If  neither  I  nor  k  is  zero,  it  intersects  curves  (1)  and  (2) 
at  no  other  points  than  their  common  points.  For  the  coordi- 
nates of  any  point  on  (1),  for  example,  but  not  on  (2),  make 
/^{x,  y)  =  0  and  /„(«,  y)  different  from  zero.  Hence  they  will  not 
satisfy  (3),  and  the  corresponding  point  cannot  be  a  point  of  (3). 

It  follows  that  if  (1)  and  (2)  have  no  points  in  common,  (3) 
intersects  neither  (1)  nor  (2).  If  we  treat  (1)  and  (2)  apart  from 
possible  geometrical  interpretation,  however,  it  is  evident  that  the 
imaginary  solutions  of  (1)  and  (2)  are  solutions  of  (3). 

By  assigning  different  values  to  I  and  k  we  may  make  (3)  satisfy 
another  condition,  as  will  be  illustrated  in  the  following  examples : 

Ex.  1.    Find  the  equation  of  the  straight  line  passing  through  the  point  of 

intersection  of  the  lines 
Y 

2x  +  y-l  =  0  (1) 

and  x  +  2y  -3  =  0  (2) 

and  the  point  (1,  2). 

l{2x+7j-l)  +  k{x  +  27j-3)  =  0     (3) 

passes  through  the  point  of  intersection  of 

(1)  and  (2),  and  is  the  equation  of  a  straight 

line,   since   it   is   an   equation   of   the   first 

degree.    Since  (3)  is  to  pass  through  the  point 

(1,  2),   (1,  2)  must  satisfy  (.3).     Therefore 

i  (2  +  2  -  1)  +  fc  (1  +  4  -  3)  =  0,  or  3  ;  +  2  /<:  =  0.     Therefore,    if   we   substitute 

k  =  -  ^lin  (3)  and  simplify,  we  shall  have  the  equation  of  the  required  line. 

It  is  found  tobex-4y  +  7  =  0  (fig.  110). 


SYSTEMS  OF  CUKVES 


173 


Ex.  2.    Find  the  equation  of  a  straiglit  line  passing  through  the  point  of 
intersection  of  the  lines 


and 

and  parallel  to  the  line 


x-2y  +  1  =  0 
x-3y  +  3  =  0, 
2x  +  3y  +  8  =  0. 


As  in  Ex.  1,  the  required  line  is 

lix  -  2y  +  1)  +  k{x  -  3y  +  3)  =  0, 
which  may  be  written 

{I  +  k)x  +  (-  21  -  Sk)y  +  {I  +  3k)  =  0. 


Since  this  line  is  to  be  parallel  to  (3), 


.     l  +  k 


2l-3k 


(1) 
(2) 
(3) 


(4) 


(§  28,  2),   whence 


2  3 

*  =  —  I  ^    Substituting  this  value  of  k  in  (4)  and  simplifying,  we  have  as  our 
required  line  2a;  +  3y-12  =  0  (fig.  111). 


Fig.  Ill 


Both  of  these  examples  could  also  have  been  solved  by  finding  the  point  of 
intersection  of  the  given  lines  and  then,  by  the  methods  of  Chap.  Ill,  passing 
the  line  through  the  point  subject  to  the  given  condition. 

93.  In  the  two  examples  of  the  last  article  both  equations  were 
of  the  first  degree.  In  this  article  we  will  solve  some  examples 
in  which  one  or  both  equations  are  of  the  second  degree. 


174 


INTERSECTION  OF  CURVES 


Ex.  1.    Find  the  equation  of  a  circle  determined  by  the  points  of  intersec- 
tion of  the  straiglit  line 

2x-y-6  =  0     (1) 
and  the  circle 

x2  +  y2-6x-6y  -7  =  0     (2) 

and  the  point  (1,  -  1)  (fig.  112). 
The  equation 

l{2x-y-6) 

+A;(x2+2/2-6x-6y-7)  =  0  (3) 

is  the  equation  of  a  circle,  since  the 
coefficients  of  x^  and  y^  are  equal ; 
and  since  it  passes  through  the  points 
of  intersection  of  (1)  and  (2),  it  only 
-piQ  112  remains  to  choose  I  and  k  so  that  it 

shall  pass  through  the  point  (1,  —1). 

3  I 
Substituting  (1,  —  1)  in  (3),  we  have  3  Z  +  5  fc  =  0,  whence  k  = Accordingly, 

the  equation  (3)  of  the  required  circle,  in  simplified  form,  is 
3  x2  +  3  2/2  -  28  X  -  13  ?/  +  9  =  0. 


Ex.  2.    Find  an  equation  representing  the  system  of  circles  passing  through 
the  points  of  intersection  of  the  circles 

x2  +  2/2  _  9  =  0 

and        x2  +  j/2  -  4x  -  2  2/  -  11  =  0 
(fig.  113). 

The  equation 

Z(x2  +  2/2-  9) 

+  A:(x2+2/2-4x-2  2/-  11)=0 

is  the  required  equation,  for  by  its  form  it  is 
the  equation  of  a  circle,  and  passes  through 
the  two  points  common  to  (1)  and  (2).  By 
assigning  different  values  to  I  and  k  we  can 
make  (3)  represent  any,  and  hence  every, 
circle  passing  through  the  common  points  of 
(1)  and  (2).    In  other  words,  it  represents  the  required  system  of  circles. 

In  particular,  if  I  and  k  are  assigned  such  values  as  to  make  the  coefficients 
of  x'  and  y^  vanish,  i.e.  k  =  —  I  in  this  example,  the  equation  reduces  to 


Fig. 113 


2x  +  2/  +  l  =  0. 


(4) 


But  this  is  the  equation  of  a  straight  line,  and  since  it  must,  from  the  way  in 
which  it  was  formed,  pass  through  the  points  common  to  the  two  circles,  it 
must  be  the  equation  of  their  common  chord. 


PROBLEMS  175 

In  general,  if            x^  +  y^  +  2  GiX  +  2Fiy  +  Ci  =  0,  (6) 

x2  +  y2  ^.  2  Ggx  +  2  F^y  +  Ca  =  0,  (6) 
are  the  equations  of  any  two  circles,  we  derive  the  third  equation 

2{Gi-G2)x  +  2{Fi-F2)y  +  {Ci-C2)  =  0  '      (7) 

by  assuming  k  =  —  I. 

If  the  circles  intersect,  (7)  is  the  equation  of  their  common  chord ;  but  if 
they  do  not  intersect,  (7)  is  called  their  radical  axis.  It  may  easily  be  proved 
to  be  perpendicular  to  the  line  of  centers  and  is  the  locus  of  points  from 
which  equal  tangents,  one  to  each  circle,  may  be  drawn. 

PROBLEMS 

Find  where  and  how  the  straight  line  intersects  the  curve  of  the  second 
degree  in  each  of  the  following  cases : 

1.  2x  +  3y  =  5,  4x2  +  92/2  +  16x-182/-ll  =  0. 

2.  X  -  y  +  1  =  0,  (X  +  2)2  _  4  2/  =  0. 

3.  x-22/  +  4  =  0,  2x2-?/2  +  8x  +  22/  +  13  =  0. 

4.  2/ -  2x  =  0,  x2  +  ^2  _  a;  +  3y  =  0. 

5.  X  -  2  2/  +  4  =  0,  5x2  -  4  2/2  +  20  =  0. 

6.  2/  =  8x-5,  2x2 +  X2/-3y2 +  6x4-42/ +  4  =  0. 

7.  2x  +  32/-6  =  0,  x2  +  4  2/2-4  =  0. 

8.  X  +  2/  -  4  =  0,  x2  -  2  X2/  +  2/2  -  20  =  0. 

9.  Find  the  length  of  the  chord  of  the  circle  x2  +  2/2  +  8x  —  42/  +  10  =  0 
cut  from  the  line  2x  —  Sy  +  S  =  0. 

10.  Find  the  tangent  to  the  curve  x2  +  6x-22/  +  5  =  0  with  slope  2. 

11.  For  what  value  of  p  will  the  parabola  y^  =  ipx  be  tangent  to  the  line 
2/-3x  +  l  =  0? 

12.  Find  the  tangents  to  the  ellipse  4  x2  +  9  7/2  =  36  which  are  parallel  to  the 
line  joining  the  positive  ends  of  the  axes. 

13.  Find  the  tangent  to  the  curve  b^x^  +  a^y^  +  2  a62x  =  0  perpendicular  to 
the  line  ax  +  by  =  ab. 

14.  Prove  that  the  line  y  =  -mx  +  2c  Vm.  is  always  tangent  to  the  hyper- 
bola xy  =  c2,  and  that  the  point  of  contact  is  ( —^ '    c  Vmj  • 

15.  Find  the  point  of  contact  of  tlie  tangent  to  the  curve 

x2_4y2  +  2x2/-2x  +  42/  =  0  with  slope  ^. 

16.  Find  the  points  of  intersection  of  the  line  82/  -  26x  =  0  and  the  curve 
x22/2  +  36  =  4  2/2. 

17.  Find  the  points  of  intersection  of  the  line  2/  =  2x  -  3  and  the  curve 
42/2  =  (x  +  3)(2x- 3)2. 


176  INTEKSECTION  OF  CURVES 

18.  Find  the  points  of  intersection  of  the  line  x  —  2y  +  2  —  0  and  the  cissoid 
X  (x2  +  2/2)  =  4  2/2. 

19.  Find   the  points  of   intersection  of   the   line   x  =  2y  and   the  curve 
16  2/2  =  4  X*  -  a*. 

20.  Find  the  points  of  intersection  of  the  line  y  =  2  x  —  2  and  the  cissoid 
a;(x2  +  2/2)  =  4  2/2. 

21.  Find  the  points  of  intersection  of  the  line  y  =  mx  and  the  cissoid 
x(x2  +  2/2)  =  2a2/2. 

22.  Find  the  points  of  intersection  of  the  line  x  —  y  —  1  =  0  and  the  witch 


y 


x2  +  4 


Find  tlie  points  of  intersection  of  the  following  pairs  of  curves : 

23.  42/2  =  x2(x  +  1),  2/2  =  x(x  +  1)2. 

24.  2/2  =  12  x,  ^2  =  (a;  +  2)  (X  -  3)2. 

25.  x2  =  y^{y  +  2),  x2  =  (2/  -  1)2(2/  +  1). 

26.  Find  the  points  of  intersection  of  the  parabolas  2/2  =  4  ax  +  4  a2  and 
2/2=-46x  +  4  62. 

27.  Find  the  points  of  intersection  of  the  parabola  x2  =  4a2/  and  the  witch 
8a3 

y  = 


X2  +  4  o2 

28.  Find  the  points  of  intersection  of   the  cissoid  y^  = and  the 

parabola  2/2  =  4  ax.  a  —  x 

29.  Find  the  poiaits  of  intersection  of  the  cissoid  y^  = and  the  circle 

x2  +  2/2-4ax  =  0.  2a-x 

30.  Find  the  points  of  intersection  of  the  hyperbola  xy  =  2aP-  and  the  witch 
_      %d? 

^~x2  +  4a2" 

31.  Find  the  points  of  intersection  of  the  witch  y  = and  the  cissoid 

4«3  *^  +  4a2 

X2  = 

5a  —  42/ 

32.  Find  the  points  of  intersection  of  the  circle  x2  +  2/2=5  a2  and  the  witch 


x2  +  4  a2 

33.  Find  the  equation  of  a  straight  line  through  the  point  of  intersection 
of  7x  -  2/  -  18  =  0  (1)  and  x  -  3  2/  -  14  =  0  (2)  and  the  point  (-  2,  1),  without 
finding  the  point  of  interaection  of  (1)  and  (2). 

34.  Find  the  equation  of  a  straight  line  through  the  point  of  intei-section  of 
2  X  -  y  +  5  =  0  (1)  and  x  -  42/  + 13  =  0  (2)  and  parallel  to  the  line  2x  +  62/  +  2  =  0, 
without  finding  the  point  of  intersection  of  (1)  and  (2). 


PROBLEMS  177 

35.  Find  the  equation  of  a  straight  line  through  the  point  of  intersection 
of  4x  -  6?/  -  5  =  0  (1)  and  G  a;  -  4?/  -  5  =  0  (2)  and  perpendicular  to  the  line 
X  —  Sy  +  1  =  0,  without  finding  the  point  of  intersection  of  (1)  and  (2). 

36.  A  circle  passes  through  the  origin  of  coordinates  and  the  points  of 
intersection  of  the  circle  x^  +  y^  =  14  and  the  line  2x  +  Sy  +  b  =  0.  Find 
its  equation. 

37.  Prove  that  (1,  1)  is  a  point  of  the  common  chord  of  the  two  circles 
a;2  +  2/^  —  4  X  =  0  and  x^  +  |/2  _  4  y  =  0. 

38.  Find  the  circle  passing  through  (1,  —  3)  and  the  points  of  intersection 
of  the  two  circles  x2  +  2/2  —  4x  —  4^/  —  8  =  0  and  x^  +  y^  +  x  +  y  —  4  =  0. 

39.  Find  a  curve  of  the  second  degree  passing  through  (1,  1)  and  the  points 
of  intersection  of  the  curves  Sx^  +  5y'^  —  15  =  0  and  2x^  —  3y^  —  G  =  0,  and 
tell  what  kind  of  a  curve  it  is. 

40.  Prove  that  a  parabola  can  be  passed  through  the  points  of  intersection 
of  the  curves  x^  -  2y^  +  x +  2y  +  1  =  0  and  3x2  +  4  2/2_2x-2  =  0. 

41.  The  center  of  a  circle  is  at  the  vertex  ^  of  a  parabola  y^  =  4px,  and  its 
diameter  is  3  J.F,  i^  being  the  focus  of  the  parabola.  Prove  that  their  common 
chord  bisects  AF. 

42.  Show  that  the  circle  described  on  any  focal  radius  of  a  parabola  as 
diameter  is  tangent  to  the  tangent  at  the  vertex  of  the  parabola. 

43.  Show  that  the  circle  described  on  any  focal  chord  of  a  parabola  as 
a  diameter  is  tangent  to  the  directrix  of  the  parabola. 

44.  If  a  circle  is  described  from  a  focus  of  an  hyperbola  as  center,  with  its 
radius  equal  to  half  the  conjugate  axis,  prove  that  it  will  touch  the  asymptotes 
at  the  points  where  they  intersect  the  corresponding  directrix. 


CHAPTER  IX 

DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

94.  Theorems  on  limits.  In  operations  with  limits  the  follow- 
ing propositions  are  of  importance. 

1.  The  limit  of  the  sum  of  a  finite  number  of  variables  is  equal 
to  the  sum  of  the  limits  of  the  variables. 

We  will  prove  the  theorem  for  three  variables ;  the  proof  is 
easily  extended  to  any  number  of  variables. 

Let  X,  Y,  and  Z  be  three  variables,  such  that  LimX  =  J, 
Lim  Y  =  B,  JAm  Z  =  C.  From  the  definition  of  Hmit  (§  53)  we 
may  write  X  =  A+a,  Y  =  B  +  b,  Z=C  +  c,  where  a,  b,  and  c  are 
three  quantities  each  of  which  becomes  and  remains  numerically 
less  than  any  assigned  quantity  as  the  variables  approach  their 
limits. 

Adding,  we  have 

X+Y-\-Z  =  A+B  +  C+a  +  b  +  c. 

Now  if  e  is  any  assigned  quantity,  however  small,  we  may 
make  a,  b,  and  c  each  numerically  less  than  -  >  so  that  a  +  b  +  c  is 

numerically  less  than  e.    Then  the  difference  between  X+Y+Z 
and  A+B  +  C  becomes  and  remains  less  than  e,  that  is, 

lim  {X+  Y  +  Z)  =  A  +  B+C=  Lim  X+  Lim  Y+  Lim  Z. 

2.  The  limit  of  the  product  of  a  finite  number  of  variables  is 
equal  to  the  product  of  the  limits  of  the  variables. 

Consider  first  two  variables  X  and  Y  such  that  Lim  X  =  A  and 
Lim  Y  =  B.    As  before,  we  have  X  =  A  +  a  and  Y  =  B  -i-b.    Hence 

XY  =  AB  +  bA  +  aB  +  ab. 
178 


THEOREMS  ON  LIMITS  179 

Now  we  may  make  a  and  h  so  small  that  hA,  aB,  and  ah  are 

e 

3 
Lim  XY  =  AB  =  (Lim  X)  (Lim  Y). 


each  less  than  - .  where  e  is  any  assigned  quantity,  no  matter  how 
small.    Hence 


Consider  now  three  variables  X,  Y,  Z.    Place  XY=  U.    Then, 

as  iust  proved, 

UmUZ  =  {UmU){lAmZ); 

that  is,  lim  XYZ  =  (Lim  XY)  (Lim  Z) 

=  (Lim  X)  (Lim  Y)  (Lim  Z). 

Similarly  the  theorem  may  be  proved  for  any  finite  number  of 
variables. 

3.  The  limit  of  a  constant  times  a  variable  is  equal  to  the  con- 
stant times  the  limit  of  the  variable. 

The  proof  is  left  for  the  student. 

4.  The  limit  of  the  quotient  of  two  variables  is  equal  to  the  quo- 
tient of  the  limits  of  the  variables,  provided  the  limit  of  the  divisor 
is  not  zero. 

Let  X  and  Y  be  two  variables  such  that  LimX  =  ^  and 
limY^B.    Then,  as  before,  X  =  A  +  a,  Y  =  B-{-b. 

„  X      A  +  a  ,      X      A      A  +  a      A       aB  —  bA 

Hence    —  = >     and = r- =  — :; — ; —  • 

Y      B  +  b  Y      B      B  +  b      B      B^  +  bB 

Now  the  fraction  on  the  right  of  this  equation  may  be  made 

less  than  any  assigned  quantity  by  taking  a  and  b  sufficiently 

small. 

TT  ^.     X      A      Lim  X 

Hence  Lim  —  =  —  = 

Y      B       LimF 

The  proof  assumes  that  B  is  not  zero. 

95.  Theorems  on  derivatives.  The  definitions  of  increment, 
continuit} ,  and  derivative  given  in  Chap.  V  are  perfectly  general, 
although  they  are  there  applied  only  to   algebraic  polynomials. 


180    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

In  order  to  extend  the  process  of  differentiation  to  other  func- 
tions, we  shaU.  need  the  following  theorems : 

1.  The  derivative  of  a  function  plus  a  constant  is  equal  to  the 
derivative  of  the  function. 

Let  w  be  a  function  of  x  which  can  be  differentiated,  let  c  be  a 
constant,  and  place  y  —  ^jl.  c 

Then  if  x  is  increased  by  an  increment  Ax,  u  is  increased  by 
an  increment  Au,  and  c  is  unchanged.  Hence  the  value  of  y 
becomes  u  +  Au  +  c. 

Whence  Ay  =  (w  +  Au  +  c)  —  {u  +  c)  =  Au. 

Therefore  ^  =  ^' 

Ax      Ax 

and,  taking  the  limit  of  each  side  of  this  equation, 

dy  _  du 

dx      dx 
Ex.   2/ =  4x3 +  3, 

^  =  1(4x3)  =  12x2. 
dx      dx 

2.  The  derivative  of  a  constant  times  a  function  is  equal  to  the 
constant  times  the  derivative  of  the  function. 

Let  -JA  be  a  function  of  x  which  can  be  differentiated,  let  c  be  a 
constant,  and  place  _  ^^^ 

Give  X  an  increment  Ax,  and  let  Au  and  Ay  be  the  correspond- 
ing increments  of  u  and  y.    Then 

Ay  =  c(u-{-  Au)  —  cu  =  cAu. 


(by  theorem  3,  §  94) 


Hence 

Ay  _    Au 
Ax        Ax 

and 

, .     Ay         -r  •     Aw 
Lim  — ^  =  c  Lim  -— 

Ax                Ax 

Therefore 

dy         du 
-^  =  c  —  > 
dx        dx 

by  the  definition  of  a  derivative. 

THEOREMS  ON  DERIVATIVES  181 

Ex.    2/  =  5(x3  +  3x2  +  l), 

dv         d 

-^  =  5--(x3  +  3x2  +  1)  =  5(3x2  +  6x)  =  lo(x2  +  2x). 

uX  uX 

3.  The  derivative  of  the  sum  of  a  fnite  number  of  functions 
is  equal  to  the  sum  of  the  derivatives  of  the  functions. 

Let  u,  V,  and  w  be  three  functions  of  x  which  can  be  differen- 
tiated, and  let  ,       , 

y  =  u  -{-  V  +  w. 

Give  X  an  increment  Ax,  and  let  the  corresponding  increments 
of  u,  V,  w,  and  y  be  Aw,  Av,  Aw,  and  Ay.    Then 

Ay  =  [u  +  A'2<-  +  V  +  Av  +  w  +  A'Z^;)  —  [u  +  v  +  w) 

=  Ait  +  A^  +  Avj ; 

,  Ay      Aw      Av      Aw 

whence  t^  =  7 — ^-r~  +  -i;—' 

Ax      Ax       Ax       Ax 

Now  let  Ax  approach  zero.    By  theorem  1,  §  94, 

-r .     Ay      _ .     Aw      -r .     A«7      -r  •     Aw . 
Lim  — ^  =  Lim f-  Lim 1-  Lim  --—  > 

Ax  Ax  Ax  Ax 

that  is,  by  the  definition  of  a  derivative, 
dy  _  du      dv      dw 

(A/*A/  tl/t/j-  (A/t/y  Ct'X' 

The  proof  is  evidently  applicable  to  any  finite  number  of 
functions. 

Ex.    ?/  =  x*- 3x3 +  2x2-7  X, 

^  =  4x3-9x2  +  4x-7. 
dx 

4.  The  derivative  of  the  product  of  a.  finite  member  of  functions 
is  equal  to  the  sum  of  the  products  obtained  by  multiplying  the 
derivative  of  each  factor  by  all  the  other  factors. 

Let  u  and  v  be  two  functions  of  x  which  can  be  differentiated, 

^^^let  y=^uv. 


182    DIFFERE^^TIATION  OF  ALGEBRAIC  FUNCTIONS 

Give  X  an  increment  Aa;,  and  let  the  corresponding  increments 
of  u,  V,  and  y  be  Aw,  Av,  and  Ay. 

Then  Ly  =  (w  +  Aw)  (v  +  At-)  —  wv 

=  w  Av  +  ■?;  Aw  +  Azt  •  Av 

,  Aw         Av         Aw  .  Aw     . 

and  —^  =  w-— +?;—-  +  -— -Av. 

A;r         ^x        Ax      Aa; 

If  now  Aic  approaches  zero,  we  have 

,.     Aw          ^.     Av   .      -r-     Aw   ,   -r-     Aw    _.      . 
Lim  — ^  =  w  Lim \-  v  Lim h  Lim  -—  •  Lim  Av. 

A«  Aa;  o^x  Ax        ,(.  q -^, 

T^  ^  T  •     Aw      i^w  ^ .     Aw      du   T .     Av      dv        ,  -r  •     a         a 
But  Lim  —^  =  -f-,  Lim  —  =  — -  >  Lim  -—  =  --  >  and  Lim  A?;  =  0 ; 
Ax     dx  Ax      dx  Ax      dx 

^,      .„  dy         dv   ,      du 

thereiore  -^  =  w  — — I-  ^  -7-  • 

dx         dx         dx 

Again,  let  y  =  uvw. 

Eegarding  uv  as  one  function  and  applying  the  result  already 
obtained,  we  have 

dy  dw  d  (uv) 

-^  =  uv  — — I-  w  —^ — - 
ax  dx  dx 


dw  . 

=  uv  — — I-  w 
dx 


[dv        dul 
dx        dxj 


dw  dv  du 

=  uv— — f-  uw  -—  +  vw-—- 
dx  dx  dx 

The  proof  is  clearly  applicable  to  any  finite  number  of  factors. 

Ex.    y  =  (3z-5)(x2  +  l)ic3, 

^  =  (3x  -  5) (x^  +  D^-i^  +  (3x  -  6).3^(?!_±i)  +  (,.  +  i),3l(^^^ 
dx      ^  '^  '   dx       ^  '  dx  ^  '  dx 

=  (3x  -  5)(x2  +  l)(3x2)  +  (3x  -  5)x3(2x)  +  (x^  +  l)x3(3) 
=  (18x3  -  25x2  +  12x  -  15)x2. 

5.  The  derivative  of  a  fraction  is  equal  to  the  denominator 
times  the  derivative  of  the  numerator  minus  the  numerator  times 
the  derivative  of  the  denominator,  all  divided  by  the  square  of 
the  denominator. 


THEOREMS  ON  DERIVATIVES  183 

l^et  y where  u  and  v  are  two  functions  of  x  which  can  be 

v 

differentiated.    Let  Aic,  Ai*,  Av,  and  Ay  be  as  usual.    Then 

.     _u-\-  Au      ti  _v  Ail  —  u  Av 
V  +  Av       V         v^-{-  V  Av 

and 


Au         Av 
Ay         Ax         Ax 


Ax         v^+  V  Av 
Now  let  Ax  approach  zero.    By  §  94, 


-r .     Au         -, .     Av 

V  Lim u  Lim  — 

Ay  Ax  Ax 

Lim-^—  = 7-, T"- — 1 

Ax  V  +  V  Lim  Av 


dy 

du         dv 
dx         dx 

whence 

dx 

v' 

Ex.    y  — , 

"      x2  +  1 

dy      (a;2  +  l)(2x)- 

-(x2-l)2a; 

1)2 

4x 

dx                     (x2  + 

(X2  +  1)2 

^.  If  y  is  a  function  of  x,  then  x  is  a  function  of  y,  and  the 
derivative  of  x  with  respect  to  y  is  the  reciprocal  of  the  derivative 
of  y  with  respect  to  x. 

Let  Ax  and  Ay  be  corresponding  increments  of  x  and  y.    Then 
Ax       1 
Ay^Ay' 
Ax 

whence  lim  -r—  = 


Ax 

dx       1 

that  is,  ~r  =  t"  ' 

dy      dy 

dx 

7.  If  y  is  a  function  of  u  and  ic  is  a  function  of  x,  then  y  is 
a  fu7iction  of  x,  and  the  derivative  of  y  with  respect  to  x  is  equal 
to  the  derivative  of  y  with  respect  to  u  times  the  derivative  of  u 
with  respect  to  x. 


184     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

An  increment  Ax  determines  an  increment  Au,  and  this  in  turn 
determines  an  increment  Ay.    Then  evidently 

Ay  _  Ay    Au 

Ax      Au    Ax 


whence 

Lim  — ^  =  lim  — ^  • 

Ax             Au 

Lim  -—  > 

Ax 

that  is, 

dy  _  dy    du 
dx      du    dx 

Ex.    y  = 

=  W2 

+  3tt 

+  1 

,    where    u  =  —, 

dy  _ 
dx~ 

=  (2 

u  +  3) 

'(- 

2\          2  +  3x2    2  _ 

X3/  ~                X2            X8  ~ 

4  +  6x2 

X5 

The  same  result  is  obtained  by  substituting  in  tlie  expression  for  y  the  value 
of  u  in  terms  of  x,  and  then  differentiating. 

96.  Formulas.    The  formulas  proved  in  the  previous  article  are : 

d(u  +  c)  _du 

(2) 
(3) 
(4) 


(5) 
(6) 

(7) 

(8) 


dx           dx 

d(cu)         du 
dx           dx 

d(u  +  v)      du      dv 
dx           dx      dx 

d{uv)         dv  1      du 
dx            dx        dx 

,  /u\         du         dv 
\v/          dx         dx 

dx    ^          v^ 

dx       1 

dy~  dy' 

dx 

dy  _  dy    du 
dx      du    dx 

dy 

dy_du 
dx~  dx' 

du 

brmula  (8)  is 

a  combination  of  (6)  and  (7), 

DERIVATIVE  OF   U""  185 

97.  Derivative  of  u".    If  tt  is  any  function  of  x  wliich  can  be 
differentiated  and  n  is  any  real  constant,  then 

—- — -  =  nu      —  • 

To  prove  this  formula  we  shall  distinguish  four  cases : 
1.  When  ri  is  a  positive  integer. 

d{u'')  _  divJ')    du 


dx  du      dx 

„_,  du 


(by  (7),  §  96) 


m.»-^-.  (by(l),  §58) 

2.  When  %  is  a  positive  rational  fraction. 

P 
Let  n  =  —  where  p  and  q  are  positive  integers,  and  place 

p 
y  =  u\ 

By  raising  both  sides  of  this  equation  to  the  g-th  power  we  have 
?/«  =  w". 

Here  we  have  two  functions  of  x  which  are  equal  for  all  values 
of  X.    If  we  give  x  an  increment  Aic,  we  have 

A(y'')  =  A(wa 
A(y^)_A(^0. 

Hx  Ax 

and  therefore  -    "^     —    \      > 

dx  dx 

whence  QV^'^  —  =  pW'^  — - » 

dx  dx 

since  ^  and  q  are  positive  integers.    Substituting  the  value  of  i) 
and  dividing,  we  have 

dx  q  dx  • 

Hence  in  this  case  also 

d  hi")  „   \du 

dx  dx 


186    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

3,  When  w  is  a  negative  rational  number. 

Let  n  =  —  m,  where  tw  is  a  positive  number,  and  place 

y  =z  U   ""  = 


d{u'^) 


Then                              -^  = 
dx 

dx 

m       1    du 

dx 

— 

^2m 

= 

-m-1  <^^' 

—  mu         -— 
dx 

Hence  in  this  case  also 

d{u^)  _ 
dx 

n      1  du 

dx 

(by  (5),  §  96) 
(by  1  and  2) 


4.  When  n  is  an  irrational  number. 

The  formula  is  true  in  this  case  also,  but  the  proof  will  not  be  given. 
As  a  particular  case  of  this  formula,  it  appears  that  1,  §  58,  is 
true  for  all  real  values  of  n. 

Ex.1.    y  =  (x3  +  4x2- 5a;  +  7)3, 

-^  =  3(a;3  +  4x2  -  5a;  +  7)2—  (x^  +  4x2  -  5x  +  7) 
dx  dx 

=  3(3x2  +  8x  _  5)(x3  +  4x2  _  5x  +  7)2.  (by  §  58) 

# 

Ex.  2.    y  =  ^/x2  +  —  =  x^  +  x-3, 
x3 

^  =!«■*- 3 x-"  (by  (3),  §96) 

_^ _3 


Ex.  ^.    y-{x  +  l)Vx2  +  1, 

=  (a;  +  i)[i(x2  +  i)-'.2x]  +  (x2  +  i)i 

=  ^(^  +  (x2  +  l)i 

_2x2  +  x  +  l 

Vz2  +  1 


HIGHER  DERIVATIVES  187 


dy_l  /     a;     \-  ^  d^  /    x    \ 

cto  ""  3  \x8  +  1/     da;  U*  +  1/ 

1  /x8  +  i\3  1-2x8 

_      1-2x8 
~3x3(x3  +  l)4' 

98.  Higher  derivatives.  It  has  been  noted  already  (§  62)  that 
the  derivative  of  the  derivative  of  a  function  is  called  the  second 
derivative  of  the  function.  Similarly  the  derivative  of  the  second 
derivative  is  called  the  third  derivative,  and  so  on.  The  succes- 
sive derivatives  are  commonly  indicated  by  the  following  notation. 


y  =^f(x),  the  original  function, 

dx 


-^  =  f'{x),  the  first  derivative, 


-— I  —  )  =  — =^  =f"(x),  the  second  derivative, 

ax  \dxj      dx 

—  (  — 4  )  =  -4  =f"'(x),  the  third  derivative, 

\dx  /      dx'^ 


dx  \dx 

dx' 


d"v 
•L  =  /^"^(a;),  the  wth  derivative. 


It  is  noted  in  §  22  that  /(a)  denotes  the  value  of  f{x)  when 
x  —  a.  Similarly  f{a),  f"{a),  f"'{a),  are  used  to  denote  the  values 
of  f'{x),  f"{x),  f"'{x)  respectively  when  x  =  a.  It  is  to  be  empha- 
sized that  the  differentiation  is  to  be  carried  out  before  the  sub- 
stitution of  the  value  of  x. 


Ex.   If/(x)  =  ^-^^,  find/"(0). 

,,,  ,  -x2  +  2x  +  l. 
^^''^-  (x-^  +  l)2  ' 
^„,  ,      2x8-Gx2-6x  +  2, 

^^^^  = WT^' 

.-.  /"(O)  =  2. 


188     DIFFERENTIATIOX  OF  ALGEBRAIC  FUNCTIONS 

99.  Differentiation  of  implicit  algebraic  functions.    Consider 
any  equation  of  the  form 

P^  +  I>xf~''  +  I>2t~''  +  I>zy'"^+  ■  •  •  +Pn-iy+K  =  ^>  (1) 
where  w  is  a  positive  integer,  and  where  some  or  all  of  the  coeffi- 
cients Pffl  Pi,  •  •  • ,  jt>„,  are  polynomials  in  x.  By  means  of  this  equa- 
tion, if  a  value  of  x  is  given,  values  of  y  are  determined.  For  if  a 
numerical  value  is  given  to  x,  the  coefficients  become  numerical 
and  the  equation  is  of  the  kind  discussed  in  Chap.  IV,  which  has 
been  shown  always  to  have  n  roots.  Hence  (1)  defines  y  as  a 
function  of  x.  This  is  the  most  general  form  of  an  algebraic 
function.  When  (1)  is  solved  for  y,  so  that  y  is  expressed  in 
terms  of  x,  y  is  an  explicit  algebraic  function.  When  (1)  is  not 
solved  for  y,  y  is  an  implicit  algebraic  function. 
For  example, 

^  a?  —  ^  xy  +  h  y^  —  ^  x  +  1  y  —  ^  =  ^, 

which  may  be  written 

5  2/'  +  (7  -  4  ic)  y  +  (3  a;'  -  6  a:  -  8)  =  0, 

defines  y  as  an  implicit  function  of  x. 
If  the  equation  is  solved  for  y,  giving 


-7+4a^±V209-}-64x-44ar^ 

^  = lo ' 

y  is  expressed  as  an  explicit  function  of  x. 

It  may  be  shown  by  advanced  methods  that  y  defined  by  (1) 
is  a  continuous  function  of  x  and  has  a  derivative  with  respect 
to  X.  Assuming  tliis,  it  is  possible  to  find  the  derivative  without 
solving  (1),  for  we  have  in  (1)  a  function  of  x  which  is  always 
equal  to  zero.  Hence  its  derivative  is  zero.  The  derivative  may 
be  found  by  use  of  the  formulas  of  the  previous  article,  as  shown 
in  the  examples. 

Ex.1.    Given  x2  +  y2  =  5. 

Then  djz^  +  y^)^ 

dx  ' 

dy 
that  IS,  2x  +  2y—  =0', 

dx 

whence  —  =  — . 

dx         y 


IMPLICIT  functio:n^s  189 

The  derivative  may  also  be  found  by  solving  the  equation  for  y.  Then 


y  =  ±y/b-  iC'^, 


dy  _  ^      -X     _ 

x 

dx          V5  -  x2 

y 

Ex.  2. 

Given 

y^  -xy  -1  =  0. 

Then 

d(2/3)      d(xy)_^ 
dx           dx 

Hence 

Sy^^/-xf--y  =  0 
dx        dx 

and 

dy           y 

dx      Sy^  —  X 

The  second  derivative  may  be   found  by  differentiating  the  result  thus 
obtained. 

Ex.  3.    If  a;2  +  y2  =  5,  we  have  found  ^  =  -  -. 

dx         y 


Therefore 


^  _  _  d^  /x\ 
c2  dx  \y/ 


d?y 

dx^         dx  \y, 

dy 
dx 


Ex.  4.  If  7/3  —  xy  —  1  =  0,  we  have  found 


y2 

y^  +  x^ 

yS 

dy  y 


dx     3  2/2  _  a; 


,dy         di^y'^-x) 
{Sy^-x)-f-y    '    '         V       , 
_,,  •  d^y  dx  dx 

Then  —^  = 

dx2  (3  2/2-x)2 

(3  2/2  _  x)2 

(3z/2_x)-^ yi^ 1) 

^  \S  2/2  -  X         \3  y2  -  a:       / 

(3  2/2-a;)2 

_      -2x2/ 

■"  (32/2 -x)3' 


190     DIFFERENTIATION  OF  ALGEBKAIC  FUNCTIONS 

100.  Tangents.    It  has  been  shown  in  §  59  that  the  tangent 
to  a  curve  ?/  =/(a^)  at  a  point  (x^,  y^  is 

where  ( —  |  denotes  the  value  of  ^  at  Ix,  yX  We  wiU  apply  this 
\dxji  dx 

to  some  of  the  curves  of  Chap.  VII,  obtaining  results  for  future 

reference. 

Ex.  1.    Consider  the  circle  Ax"^  +  Ay'^  ^r  2Gx  +  2Fy  +  C  =  0. 
Differentiating,  we  have 

2Ax  +  2Ay^  +  2G  +  2F—  =  Q; 
dx  dx 


dy_ 

Ax  +  G 

L6I1C6 

dx 

Ay  +  F 

Hence  the 

equation 

of  the  tangent 

.  is 

it  is, 

y- 

-yi  =  - 

Axi  +  G 

Ayi  +  F 

■^yi  -t  r 
Axix  -  Ax^  +  Ayiy  -  Ay^  +  Gx  -  Gxi  +  Fy  -  Fyi  =  0. 

This  equation  may  be  simplified  by  adding  to  it  the  identity 
Ax^  +  Ay^  +  2  Gxi  +  2  Fi/i  +  C  =  0, 
which  follows  from  the  fact  that  (xi,  y{)  is  on  the  circle.    There  results 
^xix  +  Ayxy  +  G  (x  +  Xi)  +  F{y  +  ^i)  +  C  =  0. 

This  result  is  easily  remembered  from  its  resemblance  to  the  equation  of  the 
circle. 

The  proofs  of  the  next  three  examples  are  left  to  the  student. 

Ex.  2.    The  tangent  to  the  ellipse  -  +  —  =  1  is  —  +  —  =  1, 
Ex.  3.    The  tangent  to  the  hyperbola  ^-^  =  lis?l?-^  =  l. 

a2         62  (j2  ft2 

Ex.  4.    The  tangent  to  the  parabola  j/2  =  4px  is  y^y  =  2p(x  +  Xi), 


NORMALS 

Ex.  5.    Consider  the  witch  x'h/  +  4  a^y  —  8  a^  =  0 
Differentiating,  we  have 


191 


ox  dz 


Hence  the  equation  of  the  tangent  is 


y  -Vi 


2X1^1 


;  («  -  «i) ; 


that  is, 
But 


Xi''  +  4  o2 

zly  +  4  a2y  -  4  a^y^  +  2  Xi^ix  -  3  xfyi  =  0. 

XjVi  +  4  o2yi  -  8  a3  =  0. 

Hence  the  equation  of  the  tangent  may  be  written 

2  xi^ix  +  (X  2  +  4  a2)  2/  +  8  a^yi  -  24  a^  =  0. 

Ex.  6.  In  the  same  manner  the  tangent  to  the  cissoid  x^  +  xy-  —  2  ay^  =  0  at 
the  point  (xi,  y{)  is  found  to  be 

(3  x^  +  y^)  X  +  (2  xxyi  -  4  ayi)  y-2  ay^  =  0. 

101.  Normals.  The  normal  to  a  curve  at  any  point  is  the 
straight  line  perpendicular  to  the  tangent  at  that  point.  To  find 
its  equation  first  find  the  slope  of  the  tangent  and  then  apply 
problem  3,  §  29. 

X^         w2 

Ex.  1.    For  the  ellipse h  —  =  1  the  slope  of  the  tangent  at  (Xi,  yi)  is 

62x,  "'      *' 

Hence  the  equation  of  the  normal  at  (xi,  yi)  is 


a-'yi 


ah/i 


y-yi  =  ^{x-  xi), 

which  is 

a%ix  -  b^iy  -  (a2  -  b!')xiyi  =  0. 

If  y  =  0, 

a2  -  62 

X  =  Xi  =  C'^Xi. 

a2 
Hence  in  fig.  114 
NF=OF-ON-ae-  e'^xx, 
F'N  =  F'O  +  ON=ae  +  e^Xi. 


Then 

F'N  _  a  +  exi  _  F'P-i 
'nF  ~  a-exi  ~  FPi  ' 


(§73) 


Fig.  114 


and  therefore,  by  plane  geometry,  the  angle  F'PiF  is  bisected  by  NPi ;  that  is, 
in  an  ellipse  the  normal  bisects  the  angle  between  the  focal  radii  drawn  to  the 
point  of  contact. 


192     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

102.  Maxima  and  minima.    The  discussiou  of  §  61  applies  here 
without  change. 


Ex.  1.    A  lever  with  the  fulcrum  at  one  end  A  (fig.  115)  is  to  be  used  to  lift  a 

jp  weight  w  applied  at  a  distance  a  from  the 
fulcrum  by  means  of  a  force  applied  at 
the  other  end  B.  The  lever  weighing  n 
units  per  unit  of  length,  required  the 
length  of  the  lever  that  the  force  required 
may  be  a  minimum. 

Let  X  =  AB,  the  length  of  the  lever,  6 
the  angle  it  makes  with  the  horizontal, 
and  F  the  force  applied  at  B.    Then  the 
weight  of  the  lever  is  nx,  and  may  be 
considered  as  applied  at  C,  the  middle  point  of  AB.    By  the  law  of  the  lever. 


Fig.  115 


Fx cos e  =  wa cos 6  +  nxl-)  cos 6, 


Then 


and 


Fz= 

wa 

X 

nx 

dF_ 
dx 

wa 

-I 

d^F 

2wa 

dx2 

X3 

2ioa    dF     ^       ,  d^F     . 

\l ,  —  =  0  and  >0. 

\     n       dx  da;2 


When  X 

Therefore  this  is  the  required  length. 


Ex.  2.    Light  travels  from  a  point  A  in  one  medium  to  a  point  B  in  another, 


the  two  media  being  separated  by  a  plane 
first  medium  is  Vi  and  in  the  second  v-z, 
required  the  path  in  order  that  the  time 
of  propagation  from  A  to  B  shall  be  a 
minimum. 

It  is  evident  that  the  path  must  lie  in  the 
plane  through  A  and  B  perpendicular  to  the 
plane  separating  the  two  media,  and  that  the 
path  will  be  a  straight  line  in  each  medium. 
We  have,  then,  fig.  116,  where  j\fiV  represents 
the  intersection  of  the  plane  of  the  motion 
and  the  plane  separating  the  two  media,  and 
ACB  represents  the  path. 


If  the  velocity  in  the 


Fig.  110 


MAXIMA  AND  MINIMA  193 


Let  MA  =  a,  NB  =  b,  MN  =  c,  and  MC  =  x.    Then  AC  =  VaF+1^^  and 
CB  =  V(c  —  a;)2  +  62,    xhe  time  of  propagation  from  J.  to  5  is  therefore 


_  Va2  +  a-2      V(c  -  a;)2  +  62 

,                                    dt              X                      c  —  X 
whence  —  = , 

^         Vi  Va2  +  X2         1)2  V(C  -  X)2  +  62 

d2<              a2  62 

and  —  = + 


dx^      t)i(a2  +  x2)  J      »2  [(c  -  a;)2  +  6^]^ 


dH 
Since  —  is  always  positive,  tlie  time  is  a  minimum  wlien 


Vi  VcC^  +  X2        V2  V(C  -  X)2  + 


(1) 


This  equation  may  be  solved  for  x,  but  it  is  more  instructive  to  proceed  as 
follows : 

X  MC 


Va2  +  x2      --l^' 

=  sin  ^. 

c  -  X         _  C'-ZV'  _ 

:  sin  \j/. 

V(c  -  X)2  +  62       <^B 

sin  0      Vi 

sin  f      V2 

Then  equation  (1)  is 

Now  0  is  the  angle  made  by  J.  C  with  the  normal  at  C  and  is  called  the  angle 
of  incidence,  and  xp  is  the  angle  made  by  CB  with  the  normal  at  C  and  is 
called  the  angle  of  refraction.  Hence  the  time  of  propagation  is  a  minimum 
when  the  sine  of  the  angle  of  incidence  is  to  the  sine  of  the  angle  of  refraction 
as  the  velocity  of  the  light  in  the  first  medium  is  to  the  velocity  in  the  second 
medium.    This  is,  in  fact,  the  law  according  to  which  light  is  refracted. 

A  case  of  a  maximum  or  a  minimum  value  sometimes  occurs 
when  the  derivative  is  infinite  and  consequently  discontinuous. 
Therefore  the  case  is  not  included  in  the  previous  discussions. 
In  practice  the  infinite  values  of  the  derivative  may  be  examined 
by  the  rule  of  §  61. 

Ex.  3.    2/  =  </{x  -  1)  (X  -  2)2  =  (x  -  1)^  (x  -  2)^ 

^  =  \ix-irHx-2)^  +  lix-l)Hx-2)-i 
ox      3  o 

=  1  (X  -  ir?(x  -  2)-i[(x  -  2)  +  2(x  -  1)] 

3x-4 

Sy/{x-  l)2(x-2) 


194    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 


•^  =  0  when  x  =  |,   and  changes  from  +  to  —  as  a;  passes  through  ^• 

Therefore  «  =  *  gives  a  maxi- 
mum   value    to     the    function. 

dy 

—  =  00  when  a;  =  1  or  2.    When 

dx       ^ 

X  =  1  — •  does  not  change  sign. 

dz  ^y 

When  X  =  2  —  changes  from  — 

dx,  ° 

to  + .  Then  x  =  2  gives  a  mini- 
mum value  of  the  function.  Its 
graph  is  in  fig.  117. 


for  it  is  that   ,  , 
32-       doif 


103.  Point  of  inflection. 

A  point  of  inflection  was 
defined  in  §  62  as  a  point 
at  which  the  curve  clianges 
from  being  concave  upward 
to  concave  downward,  or 
vice  versa;  and  the  condition 
changes  sign.    Hence  only  those  values  of  x  which 


d'y 


make  — ^  zero  or  infinity  need  be  considered  in  the  examination  of 
a  curve  for  points  of  inflection. 

Ex.  1.    Find  the  points  of  inflection  of  the  witch  y  = 


8a3 


dy 


By  differentiation,  —  = 


16a5x 


dx      (x-^  +  4  a2)2 


x2  +  4  a^ 
d^y  _  16a3(3x2-4a2) 
dx^~       (x2  +  4  a2)3 


It  is  evident  that  — -  =  0  if  x  =  ±  — ^  1  and  that  no  real  finite  value  of  x 

makes  — ^  infinite. 

dx2  9. 


We  have,  then,  to  consider  only  the  points  for  which  x 


48 


Writing  —  in  the  form 


«'(-^.)( 


-^,) 


Vs' 


,  we  see  that  if  x  < 


(x2  +  4  a'-^)3 

2a     d^y  ^.         ,..     ^    2a  d^y      ^ 

— -,  <  0  ;  and  if  x  >  — ~,  ~=-  >  0. 

V3    dx.'^  Vs  dx:^ 


2a 


dx2 

d^y      n    •*       2  a 
_|>0;  If -— :  <x 
dx2  V3 

Hence  the  curve  is  concave  downward  between  tlie  two  points  for  which 
X  = and  X  = respectively,  and  concave  upward  at  all  other  points. 

Then  there  are  two  points  of  inflection  (fig.  90,  §  82)  for  which  x  =  ±  — -  •    The 
ordinates  are  found  from  the  equation  to  be 


LIMIT  OF  RATIO  OF  ARC  TO  CHORD 

Ex.  2.    Examine  the  curve  y  =  (x  —  2)^  for  points  of  inflection. 
dy  _         1  d^y  2 


195 


By  differentiation, 


dx      3(x-2)^  dx^         9(x-2)S 


It  is  evident  that  —  =  oo  if  a;  =  2,  and  that  no  value  of  x  makes  —  =  0. 

d^y  '^'^  d^v  ^""^ 

If  X  <  2,  — ^  >  0 ;  and  if  x  >  2,  — ^  <  0.    Hence  the  point  for  which  x  =  2  is  a 
dx^  dx2 

point  of  inflection,  since  on  the  left 

of  that  point  the  curve  is  concave 

upward  and  on  the  right  of  that 

point  it  is  concave  downward  (fig. 

118).  The  ordinate  of  this  point  isO. 


Fig.  118 


104.  Limit  of  ratio  of  arc 
to  chord.  The  student  is  fa- 
miliar with  the  determination 

of  the  length  of  the  circumference  of  a  circle  as  the  limit  of  the 
length  of  the  perimeter  of  an  inscribed  regular  polygon.  So,  in 
general,  if  the  length  of  an  arc  of  any  curve  is  required,  a  broken 
line  connecting  the  ends  of  the  arc  is  constructed  by  drawing  a 
series  of  chords  to  the  curve  as  in  fig.  119.  Then  the  length  of 
the  curve  is  defined  as  the  limit  of  the  sum  of  the  lengths  of 
these  chords  as  each  approaches  zero,  and  as  their  number  there- 
fore increases  without  limit.  The 
.B  manner  in  which  this  limit  is  ob- 
tained is  a  question  of  the  Integral 
Calculus,  and  will  not  be  taken  up 
here. 

We  may  use  the  definition,  how- 
ever, to  find  the  limit  of  the  ratio  of 
the  length  of  an  arc  of  any  curve 
to  the  length  of  its  chord,  as  the 
length  of  the  arc  approaches  zero  as  a  limit,  i.e.  as  the  ends  of  the 
arc  approach  each  other  along  the  curve. 

Accordingly,  let  P^  and  P.  (fig.  120)  be  any  two  points  of  a  curve, 
i^T^  the  chord  joining  them,  and  F^T  and  P^T  the  tangents  to  the 
curve  at  those  points  respectively.  We  assume  that  the  arc  P^I^ 
lies  entirely  on  one  side  of  the  chord  ^^,  and  is  concave  toward 


Fig.  119 


196    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 


the  chord.    These  conditions  can  in  general  be  met  by  taking 
the  points  J^  and  i^  near  enough  together.    Then  it  foUows  from 

the  definition  that 


whence 

P^T+TP^      arc  P^J^ 


PP 


> 


PP 


>1. 


Fig.  120 


If  TH  is  the  perpendicular  from 
T  to  P^P^,  and  if  the  angles  P^P^T  and 
-^i^T  are  denoted  by  a  and  yS  respectively,  then  P^T  =  P^R  sec  a, 
and  TP^  =  RP^  sec  ^  =  (^^  -  P^R)  sec  /S. 

.-.  i^T  +  T^  =  iji^  sec  a;  +  (iji^-iji^)  sec /3 
=  P^P^  sec  yS  +  i^^  (sec  a  —  sec  /3), 


and 


P^T+TP^_  P,P^  sec  ^  +  P^R  {seca—  sec  /3) 
PP       ~ 


PP 

12 


Pi? 


=  sec  yS  +  -'—  (sec  a  —  sec  )Q). 


Now,  as  Pi  and  JP  approach  each  other  along  the  curve,  a  and  ^ 

both  approach  zero  as  a  limit,  whence  seca  and  secyS  approach 

PR 
unity  as  a  limit ;  and  since  -^~  is  always  less  than  unity,  it  fol- 

-^■'2 

PT+  IP 

lows  that  the  limit  of  — r^  is  unity. 

PP 
arc  PP  ^ 

Hence  ^—^  lies  between  unity  and  a  quantity  approaching 

?^.  arc  PP 

unity  as  a  limit,  and  therefore  the  limit  of  — -, — ^  is  unity,  i.e. 

J\P, 

the  limit  of  the  ratio  of  an  arc  to  its  chord  as  the  arc  approaches 
zero  as  a  limit  is  unity. 

105.  The  derivatives  -r-  and  -y-  •    On  any  given  curve  let  the 
as  as 

distance  from  some  fixed  initial  point  measured  along  the  curve  to 
any  point  P  be  denoted  by  s,  where  s  is  positive  if  P  lies  in  one 


DIRECTION  OF  A  CLTRVE 


197 


direction  from  the  initial  point,  and  negative  if  P  lies  in  the 
opposite  direction.  The  choice  of  the  positive  direction  is  purely 
arbitrary.  We  shall  take  as  the  posi- 
tive direction  of  the  tangent  that 
which  shows  the  positive  direction  of 
the  curve,  and  shall  denote  the  angle 
between  the  positive  direction  of  OX 
and  the  positive  direction  of  the  tan- 
gent by  4>. 

Now  for  a  fixed  curve  and  a  fixed 
initial  point,  the  position  of  a  point  P 
is  determmed  if  s  is  given.  Hence  x 
and  y,  the  coordinates  of  P,  are  func- 
tions of  s,  which  in  general  are  con- 
tinuous and  may  be  differentiated.    We  will  now  show  that 


dx 
ds 


=  cos  <f). 


-^  =  sin  6. 
ds 


Let  arc  PQ  =  As  (fig.  121),  where  P  and  Q  are  so  chosen  that  As 
is  positive.    Then  PE  =  Ax  and  RQ  =  Ay,  and 


PR        chord  PQ         PR 


Ax 

As      hiqPQ        arc  PQ      chord  P© 

chord  PQ 


cos  RPQ. 
arc  PQ 

RQ        chord  PQ 


RQ 


Ay_ 

As      arcPQ        arc  PQ       chord  P^ 

chord  PQ 


arc  PQ 


sin  RPQ. 


Taking    the    limit,    we    have,    since    Lim "_  _     =  1    and 


Lim  RPQ  =  4>, 


arc  PQ 


dx 
ds 


COS(f), 


dy        .     , 
-^  =  sin  9. 
ds 


(1) 


198    DIFFERENTIATIOK  OF  ALGEBRAIC  FUNCTIONS 
From  (1)  we  obtain  by  division 

dy 


ds      dy 
tan  <p  =  -7-  =  -7- ; 
ax      dx 

ds 


(2) 


by  (8),  §  96.    This  agrees  with  §  59. 

Again  from  (1),  by  squaring  each  equation  and  adding  them, 
we  have 


ds 


H^h- 


/dsV  /dsV 

By  multiplying  (3)  by  |  —  |    and  again  by  I  —  |  and  applying 

(7),  §  96,  we  have  \^-^/  W/ 


1  + 


\dx 


and 


(%h'- 


(4) 
(5) 


These  last  are  the  familiar  trigonometric  formulas 

1-f  tan'^^  =  sec'^^, 
cot^<^  +  1  =  cosec^<^. 

For  convenience  we  have  used 

a  figure  in  which  <f)  is  acute.    But 

as  s  increases  (f)  may  be  in  any 

V  ,   quadrant.    This  may  be  seen  on 

the  circle  of  fig.  122. 

The  student  may  verify  that 

formulas  (l)-(5)  are  true  in  all 

cases. 

106.  Velocity.    An  important  application  of  the  conception  of 

a  derivative  is  found  in  the  definitions  of  the  velocity  of  a  moving 

body. 


Fig.  122 


VELOCITY  199 

If  a  body  moves  so  that  the  space  traversed  is  proportional  to 
the  time,  the  motion  is  said  to  be  uniform,  and  the  velocity  is  the 
quotient  of  the  space  divided  by  the  time,  and  is  therefore  con- 
stant.   \i  t  represents  time,  s  the  space  traversed  in  the  time  t, 

s 
and  V  the  velocity,  then  for  uniform  motion  v  =  -•    When  the 

space  is  not  proportional  to  the  time  but  is  some  other  function 
of  it,  the  quotient  of  the  space  divided  by  the  time  is  called  the 
mean  or  average  velocity  during  the  time.  Thus  if  a  railroad 
train  goes  200  miles  in  5  hours,  the  mean  velocity  is  40  miles 
an  hour.    So,  in  general,  if  a  body  traverses  a  small  increment  of 

As 
space  As  in  a  small  increment  of  time  A^,  the  quotient  ■—  is  the 

mean  velocity  in  the  time  A^.  The  mean  velocity  depends  upon 
the  value  of  A^.  To  obtain  a  definition  of  the  velocity  at  the 
beginning  of  the  interval  A^,  we  think  of  A^,  and  consequently  of 

.    .         As 
As,  as  approaching  zero  as  a  limit,  and  take  the  limit  of  —  as  the 

velocity  v ;  that  is, 

T .     As      rfs 
V  =  Lim  — -  =  — . 
A^      dt 


We  note  that  if  •?;  >  0,  an  increase  of  time  corresponds  to  an 
increase  of  s ;  while  if  v  <  0,  an  increase  of  time  causes  a  decrease 
of  s.  Consequently,  the  velocity  is  positive  when  the  body  moves 
in  the  direction  in  which  s  is  measured,  and  negative  if  it  moves 
in  the  opposite  direction. 

Ex.  1.  If  a  body  is  thrown  up  from  tlie  eartli  with  an  initial  velocity  of  100  ft. 
per  second,  the  space  traversed,  measured  upward,  is  given  by  the  equation 

s  =  100t-  IC)  t'. 

Then         •  t>  =  ^  =  100  -  32«. 

,  at 

When  «  <  3^,  u  >0  and  when  e  >  3^,  u  <0.    Hence  the  body  rises  for  3^  seconds, 
and  then  falls.    The  highest  point  reached  is  100  (3^)  -  16(3|)2  =  156^. 

Ex.  2.  A  man  standing  on  a  wharf  20  ft.  above  the  water  pulls  in  a  rope 
attached  to  a  boat  at  the  uniform  rate  of  3  ft:  per  second.  Required  the 
velocity  with  which  the  boat  approaches  the  wharf. 


200     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 
Let  A  (fig.  123)  be  the  position  of  tlie  man  and  C  that  of  the  boat.    Let 

^B  =  /i  =  20,    AC=  s,    and   BC  =  x. 
dx 


We  wisli  to  find 


dt 


Now 


therefore 


X  =  Vs'^  -  400  ; 
X  s         ds 


ds 
dt 


dt       Vs-'  -  400  '^t 

But,  by  hypothesis,  s  is  decreasing  at  the  rate  of  3  ft.  per  second ;  therefore 
=  —  3,  and  tlie  required  expression  for  the  velocity  of  the  boat  is 


3s 


dx 


dt         y/g2  _  4y(J 


To  express  this  in  terms  of  the  time  we  need  to  know  the  value  of  s  when 
t  =  0.    Suppose  this  to  be  Sq  ;  then 


and 


s 
dx 


s,i  —  3 1. 


3s„  +  9« 


'^       Vs^j'-400-O.So«  +  9«2 


107.  Components  of  velocity.  When  a  body  moves  along  its 
path,  straight  or  curved,  from  P  to  ^  (fig.  124),  where  PQ  =  AS, 
X  changes  by  an  amount  PR  =  A«,  and  y  changes  by  an  amount 
RQ  =  Ay.    We  now  have 


Lim  'T~  —  'T-='^'  =  velocity  of  the  body 
in  its  path. 

Ax      dx 
Lim  — -  =  -J-  =  v^  =  component  of  ve- 
locity parallel  to  OA'. 

Lim  —  =  J   =  ^y  =  component  of  ve- 
locity parallel  to  O  Y. 


Fig.  124 


Otherwise  expressed,  v  represents  the  velocity  of  P,  v^  the 
velocity  of  the  projection  of  P  upon  OX,  and  v^  the  velocity  of 
the  projection  of  P  on  0  Y. 


COMPONENTS   OF  VELOCITY 


201 


dx 
dt 


By  (7),  §  96,  and  (3),  §  105, 

dx    ds 
ds    dt 

dsY 
dt)' 

V^  =  V  COS  (f), 


dy  _  dy   ds 
dt      ds    dt 


dxV     /dy''  "^ 
dt)      \dt 


whence 


v^  =  V  sin  ^, 


V-  =  V-  +  V' 


Ex.  A  man  walks  across  the  diameter  of  a  circular  courtyard  at  a  uniform 
rate.    A  lamp,  at  one  extremity  of  the  diameter  perpendicular  to  the  one  on 
which  he  walks,  throws  his  shadow 
on  the  wall.    Required  the  velocity 
of  the  shadow  along  the  wall. 

In  fig.  125  let  L  be  the  lamp, 
M  the  man,  and  S  the  shadow.  Let 
a  be  the  radius  of  the  courtyard 
and  c  the  uniform  velocity  of  the 
man.      Let    the    variable    OM  =  Xi 

dX] 
where  —     =  c.     Then  the   equation 

dt 
of  the  line  LS  is 

ax  —  Xiij  —  axi  =  0, 

and  that  of  the  circle  is 

a;2  +  ?/2  =:  (• ". 

Solving  these  equations,  we  have, 
for  the  coordinates  of  S, 


2a^x 


a-" 


Henc^- 


a2  +  a;f 


y  = 


a-  +  x{ 

a-    —    Xy 


and 


and 


dt 

dy 
di 


(a2  +  x{f 


2a^x?  dx,      ^   „ 

—  =  2  a-c  -    ^        „. ., 
dt  (a2  +  x^y 


4:a\    dxi         ,,   , 
-— =  -  2a-c- 


2axi 


v'i=  {  —  )  + 
\dt 


(a^  +  xf)-  dt  "  ~  (a;-  +  x{r 

fc-»  +  2  a2xf  +  x^ 


m-*^' 


(a2  +  X,-)* 


2  a-c 


The  requii-ed  velocity  is  ^ 

a   -r  X| 

The  above  solution  can  be  simplified  by  the  use  of  trigonometric  functions. 
See  Ex.  2,  §  1<>3. 


202    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

108.  Acceleration  and  force.  When  the  motion  of  a  body  is 
not  uniform,  the  velocity  at  the  end  of  an  interval  of  time  is  not 
the  same  as  at  the  beginning.  Let  v  be  the  velocity  at  the  begin- 
ning of  the  interval  A<,  and  ?;  +  Ai;  the  velocity  at  the  end.  Then 
the  limit  of  the  ratio  of  the  change  in  the  velocity  to  the  change 
in  time,  as  the  latter  approaches  zero  as  a  limit,  is  called  the 
acceleration  in  the  path  ;  that  is,  if  a  denotes  this  acceleration, 

_dv  _  d  (ds\  _  d?s 
"'~dt~di\di/~df' 

When  a  is  positive  an  increase  of  t  corres'ponds  to  an  increase 
of  V.  This  happens  when  the  body  moves  with  increasing  velocity 
in  the  direction  in  which  s  is  measured,  or  with  a  decreasing 
velocity  in  the  direction  opposite  to  that  in  which  s  is  measured. 

When  a  is  negative  an  increase  of  t  causes  a  decrease  of  v. 
This  happens  when  the  body  moves  with  decreasing  velocity  in 
the  direction  in  which  s  is  measured,  or  with  increasing  velocity 
in  the  direction  opposite  to  that  in  which  s  is  measured. 

The  force  F  which  acts  in  the  direction  of  the  path  of  a  moving 
body  is  measured  by  the  product  of  the  mass  rii  and  the  accelera- 
tiona.    Thus  ^^  ^^ 

.  F  —  ma  =  m  —  =  w  — r  • 
dt  df 

From  this  it  appears  that  a  force  is  considered  positive  or  nega- 
tive according  as  the  acceleration  it  produces  is  positive  or  nega- 
tive. Hence  a  force  is  positive  when  it  acts  in  the  direction  in 
which  s  is  measured,  and  negative  when  it  acts  in  the  opposite 
direction. 

Ex.    Let  s  =  A  +  Bt+\ Ct\ 

Then  v  =  B+Ct, 

a  =  C, 
and  F=  mC. 

If  So  and  Vo  denote  the  values  of  s  and  v  when  t  =  0,  -we  have,  from  the  last 


equations, 


So  =  ^,  Vo  =  B, 


and  the  original  equation  may  be  vsritten 

S=so  +  vot+  ]^ai?. 


ILLUSTRATIONIS  OF  THE  DEEIVATIVE  203 

As  a  special  case,  suppose  a  body  of  mass  m  thrown  vertically  upward  from 
a  point  h  ft.  above  the  surface  of  the  earth  with  an  initial  velocity  of  Vq  ft.  per 
second.    Then,  if  s  is  measured  upward  from  the  surface  of  the  earth,  we  have 

80  =  A,         F  =  —  mg,        a  =  —  g, 
where  g  is  the  acceleration  due  to  gravity.    Then  the  expression  becomes 


109.  Other  illustrations  of  the  derivative. 

1.  Bate  of  change.  \i  y  =  f{x),  a  change  of  Aa;  units  in  the 
value  of  X  causes  a  change  of  Ay  units  in  the  value  of  y.    Then 

A?/ 

— ^  is  the  change  in  y  per  unit  of  change  in  x ;  that  is,  the  change 

in  y  which  would  be  caused  by  the  change  of  a  unit  in  x,  if  A2/ 
were  proportional  to  Aic.    Passing  to  the  limit,  we  have 

dv 

-—  =  rate  of  change  of  y  with  respect  to  x. 

For  example,  the  velocity  of  a  moving  body  is  the  rate  of  change 
of  the  space  with  respect  to  the  time,  and  the  acceleration  is  the 
rate  of  change  of  the  velocity  with  respect  to  the  time. 

2.  Momentum.  The  momentum  of  a  moving  body  is  the  product 
of  the  mass  and  the  velocity ;  that  is,  if  M  is  the  momentum, 

M  =  mv. 

Now,  from  S  108,       F  =  m -—  =       ,    '  =  — —  • 
^  dt  dt  dt 

The  force  is  therefore  the  derivative  of  the  momentum  with 
respect  to  the  time,  or,  in  other  words,  the  rate  of  change  of  the 
momentum  with  respect  to  the  time. 

3.  Kinetic  energy.  The  kinetic  energy  of  a  moving  body  is 
equal  to  half  the  product  of  the  mass  into  the  square  of  the 
velocity;  that  is,  if  E  is  the  kinetic  energy, 

E  =  \  mv^. 

^,  dE      d(lmv^)  dv  ds  dv  dv       _, 

Then         -^- = -^ ~  =  mv-- =  m  —  -— =m-- =  F; 

ds  ds  ds  dt    as  at 

that  is,  the  force  is  the  derivative  of  the  kinetic  energ}'  with 
respect  to  the  space  traversed,  or,  in  other  words,  the  rate  of 
change  of  the  kinetic  energy  with  respect  to  the  space. 


204    DIFFEKENTIATION  OF  ALGEBEAIC  FUNCTIONS 

4.  Coefficient  of  expansion.  Let  a  substance  of  volume  ??  be  at  a 
temperature  t.  If  the  temperature  is  increased  by  A^,  the  pressure 
remaining  constant,  the  vohnne  is  increased  by  A  v.    The  change 

A?; 
per  unit  of  volume  is  then  — »  and  the  ratio  of  this  change  per 


unit  of  volume  to  the  change  in  the  temperature  is 


1  Av 
V  A^' 


The 


limit  of  this  ratio  is  called  the  coefficient  of  expansion ;  that  is, 

the  coefficient  of   expansion  equals  -  -y-  •    In  other  words,  the 

coefficient  of  expansion  is  the  rate  of  change  of  a  unit  of  volume 
with  respect  to  the  temperature. 

5.  Elasticity.    Let  a  substance  of  volume  v  be  under  a  pressure 
p.    If  the  pressure  is  increased  by  A^,  the  volume  is  increased  by 

Av 
—  £^v.    The  change  in  volume  per  unit  of  volume  is  then 

V 

The  ratio  of  this  change  per  unit  of  volume  to  the  change  in  the 

1   Av 

pressure  is —  j  and  the  limit  of  this'  is  called  the  compres- 

V  Lp 

sibility ;  that  is,  the  compressibility  is  the  rate  of  change  of  a  unit 
volume  with  respect  to  the  pressure. 

The  reciprocal  of  the  compressibihty  is  called  the  elasticity, 

wliich  is  therefore  equal  to  ~  v  —  • 

dv 


Ex.    For  a  perfect  gas  at  constant 
temperature, 

P  =  -- 


Therefore  the  elasticity  is 

dp  I      k\      k 

-v--  =  -v(-     )  =  -=p; 

dv  \     vV      V 

that  is,  the  elasticity  of  a  perfect  gas  is 
equal  to  the  pressure. 

G.  Areas.   Let2/=/(^)(fig.l26) 

Ite  any  curve,  C  a  fixed  point,  and 
F{x,  y)  a  variable  point  upon  it. 
We  shall  assume  that  P  lies  at  the  right  of  C  and  that  the  por- 
tion of  the  curve  between  C  and  P  lies  above  the  axis  of  x. 


M 
¥u..  120 


INTEGRATION 


205 


Draw  the  ordinates  BC  and  MP  and  let  A  denote  the  area 
BMFC.  Then  ^  is  a  function  of  x,  since  it  is  determined  when 
OM  =  a;  is  given.  Give  x  an  increment  A^  =  MN,  and  draw  the 
ordinate  NQ  and  the  Unes  BR  and  QS  parallel  to  OX.    Then 

BQ  =  Ai/,  MNQP  =  A  A, 

MNBP  =  MB  •  MN  =  y  Ax,       MNQS  =  NQ  •  MN  =  (y  +  Ay)  Ax. 

But,  from  the  figure, 

MNRB  <  MNQB  <  MNQS  *  ; 
that  is,  yAx<AA<{y  +  Ay)  Ax, 


whence 


y^-A-x^y^^y- 


Now  as  Ax  approaches  zero  as  a  limit,  —  approaches  — —  j  3/  is 

AA 
unchanged,  and  y  +  Ay  approaches  y.    Hence  -- —  >  which  lies  be- 

tween  y  and  y  +  Ay,  also  approaches  y ;  that  is, 

dA^ 
dx 

If  the  curve  lies  below  the  axis  of  x  (fig.  127),  and  we  place, 
as   before,  MNRB  =  y  Ax    and 
MNQS=  {y  +  Ay)  Ax,  these  areas 
are    negative.     We    shall   then 
have,  as  before,  — 


dA 
dx 


=  y, 


but  the  area  is  now  considered 
as  negative. 

110.  Integration.  In  many 
applications  of  the  calculus  the 
derivative  is  known,  and  the 
problem  presents  itself  to  find 


M        N 


R 


Q 


Fio.  127 
*  If  the  curve  runs  down  toward  the  right,  the  inequality  signs  will  be  reversed. 


206     DIFFEEENTIATION  OF  ALGEBRAIC  FUNCTI0:J^S 

the  function  which  has  that  derivative.  For  example,  it  may  be 
required  to  find  a  curve  when  its  slope  is  known,  or  to  find  the 
space  traversed  by  a  particle  with  known  velocity  or  acceleration, 
or  to  find  the  area  bounded  partly  by  a  known  curve,  or  to  find  a 
function  which  has  a  known  rate  of  change. 

The  process  by  which  a  function  is  found  from  its  derivative 
is  called  iiitejration.  Differentiation  and  integration  are  then 
inverse  processes,  as  are  addition  and  subtraction,  multiplication 
and  division,  involution  and  evolution.  The  methods  of  integra- 
tion are  in  general  complex  and  must  be  studied  later  in  the 
integral  calculus.  At  this  time  we  shall  give  some  simple  exam- 
ples where  the  integration  can  be  carried  out  by  reversing  the 
formulas  of  differentiation. 

In  the  first  place,  however,  we  must  notice  that  the  integration 
of  a  given  function  does  not  lead  to  a  unique  result.  For,  as  we 
have  seen  already  (§95), 

d  {u  +  c)  __  du 
dx  dx 

where  c  is  any  constant  whatever ;  that  is,  two  functions  which 
differ  hy  an  additive  constant  have  the  same  derivative. 

Conversely,  if  two  functions  have  the  same  derivative,  they  differ 
hy  an  additive  constant. 

_-      ,  ,  dv       du 

For  let  -r  =  -r' 

dx      dx 


Then 

dv      du  _  ^ 
dx      dx 

or 

d{v-u)  _Q 
dx 

Hence* 

V  —  M  =  c,  where  c  =  constant ; 

that  is, 

V  =ic  +  c. 

The  constant  c  cannot  be  determined  by  integration,  but  must 
be  fixed  by  the  special  conditions  of  the  problem  in  which  it 
occurs. 

*  A  proof  of  this  conclusion  will  be  given  in  the  second  volume. 


INTEGRATION 


207 


Ex.  1.    Required  the  curve  the  slope  of  which  at  any  point  is  twice  the 
abscissa  of  the  point. 
By  hypothesis, 

dy 
dx 
Therefore     y  =  x^  +  c.  (1) 


=  2x. 


Any  curve  whose  equation  can  be 
derived  from  (1)  by  giving  c  a  defi- 
nite value  satisfies  the  condition  of 
the  problem.  If  it  is  required  that 
the  curve  should  pass  through  the 
point  (2,  3),  we  have,  from  (1), 

3  =  4  +  c  ;     whence    c  =  —  1, 

and  therefore  the  equation   of   the 

curve  is  „,      ^o      . 

y  =  x^  —  I. 

But  if  it  is  required  that  the  curve 
should  pass  through  (—3,  10),  we 
have,  from  (1), 

10  =  9  +  c ;     whence    c  —  1, 

and  the  equation  is 

y  =  x^  +  -i. 

Ex.  2.    Required  the  space  traversed  by  a  particle  if  its  velocity  is  equal  to 
the  square  of  the  time. 

By  hypothesis. 


Therefore 


V  =  —  —  t-. 
dt 

s-  1 1^  +  c. 


The  constant  c  can  be  determined  if  we  Itnow  the  position  of  the  particle  at  a 
given  time.  For  instance,  if  when  t  —  0  the  particle  is  at  the  point  from  which 
8  is  measured,  we  must  have  c  =  0.  On  the  other  hand,  if  wlien  t  =  0  the  particle 
is  two  units  from  the  point  at  which  s  =  0,  we  have  c  =  2. 

Ex.  3.    Required  the  space  traversed  by  a  body  if  the  acceleration  is  propor- 
tional to  the  time. 

-nr    1-  dv       d^s       ,. 

We  have  o  =  —  =  ■ —  =  kt, 

dt      *2 


where  A;  is  a  known  constant.    Then  v  = 


ds_l 
di~2 


kt^  +  ci. 


and 


S  =  -M^  +  Cit  +  C2. 


The  constants  Ci  and  C2  can  be  determined  if  we  know  the  position  and  the 
velocity  of  the  body  at  a  given  time.  If,  for  examole,  we  know  that  when  <  =  0, 
a  =  0,  and  v  =  4,  we  have  cj  =  0,  ci  =  4. 


208    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 


Ex.  4.    Find  the  area  bounded  by  the  curve  y  =  ^  (x^  —  3x'^  —  Ox  +  27),  the 
axis  of  X,  and  the  ordinates  x  =  4  and  x  =  5. 

If  A  is  the  area  C DP M  (fig.  129),  where  OC  =  4  and  OM  =  x,  then  (§  109,  6) 


d  A        1 

—  =  -(x3-3x2-9x  +  27). 


whence 


(1) 


Therefore 


1  /x*  9 

8\4  2 


x) 

/      2 


Fig.  129 

If  X  =  4,  MP  coincides  with  CD  and  therefore  J.  =  0.    Substituting  in  (1) 
the  corresponding  values  x  =  4,  yl  =  0,  we  find  c  =  —  |. 

9 
2' 
If  X  =  5,  ^  =  CDEF.    Hence 

CZ>J?F  =  ^  (fi-l A  _  125  -  2f  A  +  135)  -  I  =  2/j. 

Ex.  6.    Find  the  area  bounded  by  the  axis  ofx  and  tlie  portion  of  the  curve 
y  =  ^  (x^  —  3  x2  —  9  X  +  27)  between  x  =  —  3  and  x  =  3. 
We  now  let  A  =  the  area  6?iVQ  (fig.  129). 


Then,  as  before. 


dA_l 

dx  ~8 


(x3  -3x2-  9x  +  27). 

1  /x*  9  \ 

^  =  -  I x3  -  -  x2  +  27  X )  + 

8\4  2  / 

When  X  =  -  3,  ^  =  0  ;  therefore  c  =  2^^ , 

and  A  =  -(-~x^-~x'^  +  21x]  +  ^^'^ 

8\4  2  ' 


32 


Placing  X  =  3,  we  have  area  GQH  —  13 J^. 


PROBLEMS  209 

PROBLEMS 


dy 
Find  -^  in  each  of  the  following  cases : 
dx 

1.  2/  =  (3a;  +  l)(x2  +  2x  +  1). 


16. 


y 


2.  y  =  (3x2  +  6x  +  1) (5x2  +  lOx  +  5).  x^  +  x2  +  1 

x  +  a  Vx2  +  1 


4.  y: 

5.  y 

6.  y 


x  +  a 

x3  +  l 

X3-1 

2x2- 

4x  +  3 

3x2- 

6x  +  1 

x^  —  x- 

2  +  X- 

1 

18.  2/  =  (2x  -3)2(x  +  l)3. 

19.  2/  =  (3x  -  5)2(x2  -  5x  +  1). 

20.  2/  =  (x  +  l)Va;2-  1. 

21.  1/  =  (x2  -  4x  +  3)'^(x3  +  1)3. 
2.2.  y  =  Vx  +  1  +  Vx  -  1. 


a;'*-l  23.  2/  =  x  + Vx2  +  1. 

2 


7.  2/  =  2xi  +  8x^--  +  -..  2L  y=W^^Tl  +  -^. 


V3x2  +  1 


8.  z/  =  4x2-0x  +  ^-|.  25.  2/=^(|-J, 

9.  7/  =  V^-J-.  „„                  Vx2  +  1 

Vx  26.  y  = — • 

^  X  X  —  1 

10.  y  =  ^X2  —  -^X  +  = =:•  „„                (X2  +  1)5 

Vx  ^x2               27.  y  =  ^^ ^, 

^X  vx                                                (X3  +  1)* 

11.  y  =  (3x2 -ox +  6)2.  ^ 

12.  2/  =  (x2  +  l)3.  28.  2/ 

13.  ?/  =  V4x2  +  5x-6.  -Q              

3/ —  ^~  / 

14.  y  =  Vx2  +  X  -  1.  ^02  -  x2 

15.  2/^ -J—.  30.2/ 


X  +  Vl  +  X2 
X 


x2  +  1  *  x  +  Va2  +  x2 

Find  —  from  each  of  the  following  equations : 
dx 

31.  X*  -  4x22/2  +  2/3  =  0.  34.  x5  +  2/*  -  x'  -  2/  =  0. 

32.  x5  -  2/5  -  x3  +  2/  =  0.  35.  (X  +  2/)^  +  (X  -  2/)^  =  a. 

33.  x«2/*  +  (X  -  yY  =  0.  36.  y"-  =  ^-^  • 

X     y 

Find  ^  and  —^  from  each  of  the  following  equations : 
dx  dx2 

37.  5  x2  +  2  2/2  =  10.  40.  2/'  =  ct^ (X  +  2/). 

38.  x^  +  y'  =  a^.  41_  2/'  +  2/  =  x^. 


39.  t^  +  ^  =  1 


?!  4.  tL_ 

a2      62      "•  42.  2/' -2x3 +  4x2/  =  0. 


210    DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

43.  Find  the  tangent  and  the  normal  to  the  parabola  y^  —  4y  —  6x  —  9  =  0 
at  a  point  the  abscissa  of  which  is  —  2. 

44.  Find  the  equations  of  the  tangent  and  the  normal  to  the  circle 

x^  +  y^-iz  +  6y-2i  =  0 
at  the  point  (1,  3). 

45.  Find  the  equation  of  the  tangent.to  the  witch  y  = at  the  point 

for  which  x  =  1. 

46.  Find  the  tangent  to  the  curve  x^  —  y^  +  x^  —  y  =  0  at  the  point  the 
abscissa  of  which  is  1. 

47.  Find  the  tangent  to  the  curve  x^y  +  x^  —  x^  +  y  =  0  at  the  point  the 
abscissa  of  which  is  1. 

48.  Find  the  equation  of  the  tangent  to  the  curve  y^  —  xy  —  a  =  0  at  the 
point  {xi,  Vi). 

49.  Find  the  equation  of  the  tangent  to  the  curve  x  =  y^  +  1  &t  the  point 
(xi,  yi). 

50.  Find  the  equation  of  the  tangent  to  the  curve  y-  =  x^  at  the  point  (xi,  yi). 

51.  Find  the  equations  of  the  tangent  and  the  normal  to  the  curve  y  —  x  -i — 
at  the  point  (xi,  yi). 

52.  Find  the  equation  of  the  tangent  to  the  curve  Vx  +  Vy  =  Va  at  the 
point  (Xi,  yi). 

53.  Find  the  equation  of  the  tangent  to  the  curve  x^  +  y^  —  a*  at  the  point 

(xi,  yi)- 

54.  Find  the  tangent  and  the  normal  to  the  ellipse  3x^  +  5y^  =  lo  at  the 
upper  end  of  the  ordinate  through  the  right-hand  focus. 

55.  Find  the  equations  of  the  tangent  and  the  normal  to  the  hyperbola 
4  x^  —  2/2  —  12  at  a  point  the  abscissa  of  v/hich  is  equal  to  its  ordinate. 

56.  Find  in  terms  of  x,  y,  and  —  the  projections  upon  OX  of  the  portions 

dx 
of  the  tangent  and  the  normal  between  the  point  of  contact  and  OX.    These 
are  called  the  suhtangent  and  the  subnormal. 

dy 

57.  Find  in  terms  of  x,  y,  and  —  the  lengths  of  the  portions  of  the  tangent 

dx 
included  between  the  point  of  contact  and  the  coordinate  axes. 

58.  Prove  that  a  normal  to  an  hyjjerbola  makes  equal  angles  with  the  focal 
radii  drawn  to  the  point  where  the  normal  intersects  the  hyperbola. 

59.  Prove  that  a  normal  to  a  parabola  makes  equal  angles  with  the  axis  of 
the  parabola  and  the  line  drawn  from  the  focus  to  the  point  where  the  normal 
intersects  the  parabola. 

60.  Show  that  for  an  ellipse  the  segments  of  the  normal  between  the  point 
of  the  curve  at  which  the  normal  is  drawn  and  the  axes  are  in  the  ratio  a^  :  b"^. 

61.  Find  the  point  at  which  the  tangent  to  the  curve  x^  —  xy  —  1  =  0  has 
the  slope  2. 


PROBLEMS  211 

62.  Find  the  coordinates  of  a  point  on  the  ellipse 1-  —  =  1  such  that  the 

tangent  there  is  parallel  to  the  line  joining  the  positive  extremities  of  the  major 
a,nd  the  minor  axes. 

63.  Find  a  point  on  the  ellipse  -^  +  ^  =  1  such  that  the  tangent  there  is 
equally  inclined  to  the  two  axes. 

64.  Prove  that  the  portion  of  a  tangent  to  an  hyperbola  included  by  the 
asymptotes  is  bisected  by  the  point  of  tangency. 

65.  If  any  number  of  hyperbolas  have  the  same  transverse  axis,  show  that 
tangents  to  the  hyperbolas  at  points  having  the  same  abscissa  all  pass  through 
the  same  point  oh  the  transverse  axis. 

66.  If  a  tangent  to  an  hyperbola  is  intersected  by  the  tangents  at  the  verti- 
ces in  the  points  Q  and  B,  show  that  the  circle  described  on  QR  as  a  diameter 
passes  through  the  foci. 

67.  Prove  that  the  ordinate  of  the  point  of  intersection  of  two  tangents  to  a 
parabola  is  the  arithmetical  mean  between  the  ordinates  of  the  points  of  con- 
tact of  the  tangents. 

68.  If  P,  Q,  and  R  are  three  points  on  a  parabola,  the  ordinates  of  which 
are  in  geometrical  progression,  show  that  the  tangents  at  P  and  R  meet  on  the 
ordinate  of  Q. 

69.  Show  that  the  tangents  at  the  extremities  of  the  latus  rectum*  of  a 
parabola  are  perpendicular  to  each  other. 

70.  Prove  that  the  tangents  at  the  ends  of  the  latus  rectum  of  a  parabola 
intersect  on  the  directrix. 

71.  Prove  analytically  that  if  the  normals  at  all  points  of  an  ellipse  pass 
through  the  center,  the  ellipse  is  a  circle. 

72.  Prove  that  the  tangent  at  any  point  of  the  parabola  y'^='ipx  will  meet  the 
directrix  and  the  latus  rectum  produced  in  two  points  equidistant  from  the  focus. 

73.  Find  the  length  of  the  perpendicular  from  the  focus  of  the  parabola 
y2  =  4px  to  the  tangent  at  any  point  (xi,  j/i),  in  terms  of  Xi  and  p. 

74.  If  perpendiculars  are  let  fall  on  any  tangent  to  a  parabola  from  two  given 
points  on  the  axis  which  are  equidistant  from  the  focus,  prove  that  the  difference 
of  their  squares  is  constant. 

75.  Show  that  the  product  of  the  perpendiculars  from  the  foci  of  an  ellipse 
upon  any  tangent  equals  the  square  of  half  the  minor  axis. 

76.  Find  the  equation  and  the  length  of  the  perpendicular  from  the  center 
to  any  tangent  to  the  ellipse 1-  —  =  1. 

77.  At  what  angles  t  do  the  loci  2/2-4a;-|-4=Oand2/-a;  +  l  =  0  intersect  ? 

*The  latus  rectum  of  a  conic  is  the  chord  through  the  focus  perpendicular  to 
the  axis. 

fThe  angle  between  two  curves  is  the  angle  between  their  tangents  at  their 
point  of  intersection. 


212     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

78.  Find  the  angle  between  the  straight  line  y  —  2x  —  2  and  the  cissoid 
X  {x^  +  y''^)  —  i  y'^  at  each  of  their  points  of  intersection. 

79.  At  what  angle  do  the  circles  x^  -hy-  -Q  =  0,  x^  +  y^  —  iix-6y  +  9  =  0 
intersect  ? 

80.  Prove  that  the  center  of  each  of  the  circles 

x^  +  y^  =  a2    and    x^  +  y^-2ax  =  0 
is  a  point  of  the  other,  and  find  the  angle  at  which  they  intersect. 

81.  At  what  angle  do  the  circle  x'^  +  y^  =  21  and  the  parabola  y'^  =  ix  inter- 
sect each  other  ? 

82.  Show  that  the  curves \-  —  =  1  and ^  =  1  cut  each  other  at 

right  angles  and  are  confocal.  ' 

83.  Prove  that  an  ellipse  and  an  hyperbola  with  the  same  foci  cut  each 
other  at  right  angles. 

84.  If  two  concentric  equilateral  hyperbolas  are  described,  the  axes  of  one 
being  the  asymptotes  of  the  other,  show  that  they  intei-sect  at  right  angles. 

85.  Find  the  angle  between  the  parabolas  y"^  —  iax  and  x^  =  4ay  at  each 
of  their  points  of  intersection. 

86.  Find  the  angle  between  the  parabola  x"  =  iay  and  the  witch  y  =  — 

at  each  of  their  points  of  intersection.  "^ 

87.  Prove  that  the  cissoid  y^  = and  the  parabola  y^  =  4ax  intersect 

at  right  angles  at  the  origin.  "" 

88.  Find  the  angles  of  intersection  of  the  cissoid  y-  = and  the  circle 

x2  +  y2_4ax  =  0.  2a -X 

89.  Find  the  angle  of  intersection  of  the  witch 

8  a^  4  7/8 

y  =  — ^  and  the  -cissoid  x^  — 


x'^  +  ia^  5  a  —  4  y 

90.  Find  the  angles  of  intersection  of  the  circle  x^  +  y-  =  a  a^  and  the  witch 

8a3 


y 


«2  +  4  a^ 


91.  Find  the  angle  between  the  strophoid  y  =  ±x\ and  the  circle 

0,09  \  a  +  X 

92.  Find  the  angles  of  intersection  of  the  curves 

2/2  =  2  ax    and    x^  +  y^  —  S  axy  =  0. 

93.  It  is  required  to  fence  off  a  rectangular  piece  of  ground  to  contain  a 
given  area,  one  side  to  be  bounded  by  a  wall  already  constructed.  Required 
the  dimensions  of  the  rectangle  which  will  require  the  least  amount  of  fencing. 

94.  A  man  on  one  side  of  a  river,  the  banks  of  which  are  assumed  to  be 
parallel  straight  lines  i  mi.  apart,  wishes  to  reach  a  point  on  the  opposite  side 
of  the  river  and  3  mi.  further  along  the  bank.  If  he  can  walk  4  mi.  an  hour 
and  swim  2  mi.  an  hour,  find  the  route  he  should  take  to  make  the  trip  in  the 
least  time. 


PKOBLEMS  213 

95.  A  rectangular  piece  of  land  to  contain  96  sq.  rd.  is  to  be  inclosed  by  a 
fence  and  divided  into  two  equal  parts  by  a  fence  parallel  to  one  of  the  sides. 
What  must  be  the  dimensions  of  the  rectangle  that  the  least  amount  of  fence 
may  be  required  ? 

96.  What  are  the  dimensions  of  the  rectangular  beam  of  greatest  volume 
that  can  be  cut  from  a  log  a  ft.  in  diameter  and  b  ft.  long,  assuming  the  log  to 
be  a  circular  cylinder  ? 

97.  The  hypotenuse  of  a  right  triangle  is  given.  How  shall  the  sides  be 
chosen  so  that  the  area  shall  be  a  maximum  ? 

98.  Two  towns  A  and  B  are  situated  respectively  2  mi.  and  3  mi.  back 
from  a  straight  river  from  which  they  are  to  get  their  water  supply,  both  from 
the  same  pumping  .station.  At  what  point  on  the  bank  of  the  river  should  the 
station  be  placed,  that  the  least  amount  of  piping  may  be  required,  if  the  neai'est 
points  of  the  river  to  A  and  B  respectively  are  10  mi.  apart? 

99.  AB  and  CD  are  two  parallel  lines  distant  b  units  apart.  A  transversal 
BF  is  drawn,  intersecting  the  transversal  AD  at  E.  For  what  position  of  F  is 
the  sum  of  the  ai'eas  of  the  two  triangles  AEB  and  FED  a  minimum  ? 

100.  A  right  cone  is  generated  by  revolving  an  isosceles  triangle  of  constant 
perimeter  about  its  altitude.  Show  that  the  cone  of  greatest  volume  will  be 
obtained  when  the  length  of  the  side  of  the  triangle  is  three  fourths  the  length 
of  the  base. 

101.  Into  a  full  conical  wine  glass  whose  depth  is  a  and  angle  at  the  base  is 
2  a  there  is  carefully  dropped  a  spherical  ball  of  such  size  as  to  cause  the  greatest 
overflow.    Show  that  the  radius  of  the  ball  is 

a  sin  a 


sin  a  +  cos  2  a 


102.  Two  ships  are  sailing  uniformly  with  velocities  w,  v  along  lines  inclined 
at  an-  angle  6.  Given  that  at  a  certain  time  the  ships  are  distant  respectively 
a  and  6  from  the  point  of  intersection  of  their  courses,  show  that  the  least  dis- 
tance between  the  ships  is 

(av  —  6(t)sin  0 

(u2  +  v"^  -2uv  cos  6)^ 

103.  Find  the  least  ellipse  which  can  be  described  about  a  given  rectangle, 
assuming  that  the  area  of  an  ellipse  with  semiaxes  a  and  b  is  -n-ab. 

104.  Find  what  sector  must  be  taken  out  of  a  given  circle  in  order  that  it 
may  form  the  curved  surface  of  a  cone  of  maximum  volume. 

105.  The  stiffness  of  a  rectangular  beam  varies  as  the  product  of  the  breadth 
and  the  cube  of  the  depth.  Find  the  dimensions  of  the  stiffest  rectangular  beam 
that  can  be  cut  from  a  circular  cylindrical  log  of  radius  a  in. 

106.  Tlie  strength  of  a  rectangular  beam  varies  as  the  product  of  its  breadth 
and  the  .s<juare  of  its  depth.  Find  the  dimensions  of  the  strongest  rectangular 
beam  that  can  be  cut  from  a  circular  cylindrical  log  of  radius  a  in. 


214     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

107.  The  fuel  consumed  by  a  steamship  is  proportional  to  the  cube  of  the 
velocity  which  would  be  given  to  the  steamship  in  still  water.  If  it  is  required 
to  steam  a  fixed  distance  against  a  current  flowing  a  mi.  an  hour,  find  the 
most  economical  rate. 

108.  A  cistern  in  the  form  of  a  circular  cylinder  open  at  the  top  is  to 
be  constructed  to  contain  a  given  amount.  Required  the  dimensions  that  the 
amount  of  material  expended  may  be  the  least. 

109.  Required  the  right  circular  cone  of  greatest  volume  which  can  be 
inscribed  in  a  given  sphere. 

110.  A  power  house  stands  upon  one  side  of  a  river  of  width  b  mi.  and  a 
manufacturing  plant  stands  upon  the  opposite  side  a  mi.  downstream.  Find 
the  most  economical  way  to  construct  the  connecting  cable  if  it  costs  m  dollars 
per  mile  on  land  and  n  dollars  per  mile  through  water. 

111.  Find  the  isosceles  triangle  of  greatest  area  which  can  be  cut  from  a  semi- 
circular board,  the  vertex  of  the  triangle  being  at  the  center  of  the  diameter. 

112.  Find  the  isosceles  triangle  of  greatest  area  which  can  be  placed  in  a 
figure  bounded  by  a  portion  of  a  parabola  and  a  straight  line  perpendicular  to 
the  axis  of  the  parabola,  assuming  that  the  vertex  of  the  triangle  lies  in  the 
straight  line  and  that  the  base  is  parallel  to  the  straight  line. 

113.  Find  the  point  of  inflection  of  the  curve  y  =  a  +  {b  —  x)^. 

114.  Find  the  points  of  inflection  of  the  curve  y  = 

x2  +  1 

115.  Examine  the  curve  y  =  (x  —  l)^{x  +  1)^  for  maxima  and  minima  and 
points  of  inflection. 

116.  Find  the  maximum  and  the  minimum  ordinates  and  the  points  of 

inflection  of  the  curve  y^  =  x(x^  —  a^). 

a 

117.  Find  the  points  of  inflection  of  the  curve  y  = 


X2-1-4 


118.  Show  that  the  strophoid  y  =  ±  x  \  has  no  point  of  inflection. 

\  a  +  x 

119.  Find  the  points  of  inflection  of  the  curve  a*y'^  =  a^x*  —  x^. 

120.  Find  the  points  of  inflection  of  the  curve  |-|  +  f-J  =  1. 

121.  Find  where  the  rate  of  change  of  the  ordinate  of  the  curve 

j/  =  a;3  -6a;2  +  3x  +  5 
is  equal  to  the  rate  of  change  of  the  slope  of  the  tangent. 

122.  A  body  moves  in  a  straight  line  according  to  the  law  8=  ^t*  —  At^  +  16t^. 
Find  its  velocity  and  acceleration.  When  is  it  stationary  ?  When  is  its  velocity 
a  maximum  ?    During  what  interval  is  it  moving  backward  ? 

123.  A  particle  is  moving  along  the  curve  y^  —  4x,  and  when  a;  =  4  its  ordi- 
nate is  increasing  at  the  rate  of  10  ft.  per  second.  At  what  rate  is  the  abscissa 
then  changing,  and  how  fast  is  the  particle  moving  along  the  curve  ?  Where 
will  the  abscissa  be  changing  ten  times  as  fast  as  the  ordinate  ? 


PROBLEMS  215 

124.  Two  points,  having  always  the  same  abscissa,  move  in  such  a  manner 
that  each  generates  one  of  the  curves  y  =  x^  —  12x^  +  4x  and  y  =  x^  —  8x^  —  8. 
Wlien  are  the  points  moving  with  equal  speed  in  the  direction  of  the  axis  of  y? 
What  will  be  true  of  the  tangent  lines  to  the  curves  at  these  points  ? 

12a.  The  top  of  a  ladder  a  units  long  slides  down  the  side  of  a  vertical  wall 
which  rests  on  horizontal  land.  Find  the  ratio  of  the  velocities  of  its  top  and 
bottom. 

126.  The  altitude  of  a  variable  cylinder  is  constantly  equal  to  the  diameter 
of  its  base.  If  when  the  altitude  is  6  ft.  it  is  increasing  at  the  rate  of  2  ft.  an 
hour,  how  fast  is  the  volume  increasing  at  the  same  instant  ? 

127.  A  boat  moving  8  mi.  an  hour  is  laying  a  submarine  cable.  Assuming 
that  the  water  is  100  ft.  deep,  that  the  cable  is  attached  to  the  bottom  of  the 
sea  and  stretches  in  a  straight  line  to  the  stern  of  the  boat,  at  what  rate  is  the 
cable  leaving  the  boat  when  120  ft.  have  been  paid  out  ? 

128.  A  ball  is  swung  in  a  circle  at  the  end  of  a  cord  4  ft.  long  so  as  to  make 
100  revolutions  a  minute.  If  the  cord  breaks,  allowing  the  ball  to  fly  off  at  a 
tangent,  at  what  rate  will  it  be  receding  from  the  center  of  its  previous  path 
10  sec.  after  the  cord  breaks,  if  no  allowance  is  made  for  any  new  force  acting? 

129.  A  body  slides  down  an  inclined  plane  at  such  a  rate  that  the  distance 
traversed  at  the  end  of  t  sec.  from  the  time  it  begins  to  move  is  5 1^.  If  the  plane 
is  inclined  to  the  horizon  at  an  angle  of  30°,  what  is  the  vertical  velocity  of  the 
body  at  the  end  of  3  sec.  ? 

130.  A  roll  of  boJt  leather  is  unrolled  on  a  horizontal  surface  at  the  rate  of 
5  ft.  per  second.  If  the  leather  is  ^  in.  thick  and  at  the  start  the  roll  was  2  ft. 
in  diameter,  at  what  rate  is  the  radius  decreasing  at  the  end  of  3  sec,  if  the  roll 
is  assumed  to  be  a  true  circle  ? 

131.  An  elevated  car  running  at  a  constant  elevation  of  40  ft.  above  the 
street  pas.ses  directly  over  a  surface  car,  the  tracks  of  the  two  cars  crossing  at 
right  angles.  If  the  speed  of  the  elevated  car  is  IG  mi.  per  hour  and  the  speed 
of  the  surface  car  8  mi.  per  hour,  at  what  rate  are  the  cars  separating  5  min. 
after  they  meet  ? 

132.  Find  the  curve  the  slope  of  phich  at  any  point  is  3  more  than  the 
square  of  the  abscissa  of  that  point  and  which  passes  through  the  point  (1,  —  3). 

133.  x'^ind  the  curve  the  slope  of  which  at  any  point  is  equal  to  the  square 
of  the  reciprocal  of  the  abscissa  of  the  point  and  which  passes  through  (2,  1). 

134.  Find  the  curve  the  slope  of  which  at  any  point  is  equal  to  the  square 
root  of  the  absci.ssa  of  the  point  and  which  pas.ses  through  (4,  9). 

135.  Prove  that  any  curve  the  slope  of  which  at  any  point  is  proportional 
to  the  abscissa  of  the  point  is  a  parabola. 

136.  Find  the  curve  the  slope  of  which  at  any  point  is  proportional  to  the 
square  of  the  ordinate  of  the  point  and  which  passes  through  (1,  1). 

137.  Find  the  area  of  each  arch  of  the  curve  ?/  =  150a;  -  25x2  _  ^s. 


216     DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

138.  Find  the  area  of  the  arch  of  the  curve  y  =  x^  —  Sx^  ~  dx  +  27. 

139.  Show  that  the  area  bounded  by  any  parabola  y"  —  4px,  the  axis  of  x, 
and  tlie  ordinate  tlirougli  any  point  of  tlie  curve  is  two  thirds  the  area  of  a  rec- 
tangle the  sides  of  which  are  the  coordinates  of  the  point. 

140.  Expre.ss  the  area  between  the  curve  y  =  x",  the  axis  of  x,  and  the  ordi- 
nate through  the  point  (h,  k)  of  the  curve  as  a  rational  function  of  h  and  k. 

141.  Find  the  area  of  the  three-sided  figure  bounded  by  the  coordinate  axes 
and  the  curve  x^  -f-  2/^  =  a'  (§  G9). 

142.  Find  tlie  area  between  the  parabola  y-  =  ^x  and  the  straight  line 
2  y  -  x  =  0. 

143.  Find  the  area  between  the  parabolas  y'^  =  4  ax  and  x^  =  4  ay. 

144.  Find  the  area  of  the  crescent-shaped  figure  between  the  curves 
y  =  x"^  -h  5  and  ?/  =  2  x-  -|-  1. 

145.  Find  the  area  of  the  closed  figure  bounded  by  the  curves  y'^  —  \(jx 
and  2/2  ;=  x^. 


CHAPTEE  X 


CHANGE  OF  COORDINATE  AXES 


111.  Introduction.  So  far  we  have  dealt  with  the  coordinates 
of  any  point  in  tlie  plane  on  the  supposition  that  the  axes  of  coor- 
dinates are  fixed,  and  therefore  to  a  given  point  corresponds  one, 
and  only  one,  pair  of  coordinates ;  and,  conversely,  to  any  pair  of 
coordinates  corresponds  one,  and  only  one,  point.  But  it  is  some- 
times advantageous  to  change  the  position  of  the  axes,  i.e.  to  make 
a  transformation  of  coordinates,  as  it  is  called.  In  such  a  case 
we  need  to  know  the  relations  between  the  coordinates  of  a  point 
with  respect  to  one  set  of  axes  and  the  coordinates  of  the  same 
point  with  respect  to  a  second  set  of  axes. 

The  equations  expressing  these  relations  are  called  formulas  of 
transformation.  It  must  be  borne  in  mind  that  a  transformation 
of  coordinates  never  alters  the  position  of  tlie  point  in  the  plane, 
the  coordinates  alone  being  changed  because  of  the  new  standards 
of  reference  adopted. 

112.  Change  of  origin  without  change  of  direction  of  axes. 
In  this  case  a  new  origin  is  chosen,  but  tlie  new  axes  are  respec- 
tively parallel  to  the  original  axes.  , 

Let  OX  and  or  (fig.  130)  be 
the  original  axes,  and  O'X'  and 
O'Y'  the  new  axes  intersecting 
at  0',  the  coordinates  of  0'  with 
respect  to  the  original  axes  being 
Xq  and  ^o-  Let  P  be  any  point 
in  the  plane,  its  coordinates  being 
X  and  y  with  respect  to  OX  and 
OY,  and  x'  and  y'  with  respect  to 
O'X'  and  O'Y'.  Draw  P3IM'  paral- 
lel to  OY,  intersecting  OX  and  O'X'  at  M  and  M'  respectively. 

217 


0' 


M 


N 


Fig.  130 


M' 


218  CHANGE  OF  COORDINATE  AXES 

Then  0M=  x,  MP  =  y,  0'M'=  x',  M'P  =  y',  0N=  y^,  and  N0'=  x^ 

But  OM  =  NM'  =  NO'  +  O'M', 

and  MP  =  MM'  +  M'P  =  0N+  M'P. 

.-.  x  =  x^+  x',    y  =  yo  +  y\ 

whicli  are  the  required  formulas  of  transformation. 

Ex.  1.  The  coordinates  of  a  certain  point  are  (3,  -  2).  What  will  be  the 
coordinates  of  this  same  point  with  respect  to  a  new  set  of  axes  parallel  respec- 
tively to  the  first  set  and  intersecting  at  (1,  —  1)  with  respect  to  OX  and  OY  ? 

Here  a;o  =  1,  yo  =  —  1,  x  =  3,  and  y  =  —  2.  Therefore  3  =  1  +  x'  and 
—  2  =  —  1  +  y',  whence  x'  =  2  and  y'  —  —\. 

Ex.  2.  Transform  the  equation  i/^  —  2?/  —  3x  —  5  =  0toa  new  set  of  axes 
parallel  respectively  to  the  original  axes  and  intersecting  at  the  point  (—  2,  1). 

The  formulas  of  transformation  are  x  =  —  2  +  x',  y  =  \-\-  y".  Therefore 
the  equation  becomes 

(1  +  y')2  _  2(1 -I- 2/0  -  3(- 2  +  X')  -  5  =  0, 
or  2/'2-3x'  =  0. 

Ai  no  point  of  the  curve  has  been  moved  in  the  plane  by  this  transformation, 
the  curve  has  been  changed  in  no  way  whatever.  Its  equation  is  different  because 
it  is  referred  to  new  axes. 

After  the  work  of  transformation  has  been  completed  the  primes  may  be 
dropped.  Accordingly,  the  equation  of  this  example  may  be  written  y^  —  Sx  =  0, 
or  y^  =  S  X,  the  new  axes  being  now  the  only  ones  considered. 

113.  One  important  use  of  transformation  of  coordinates  is 
the  simplification  of  the  equation  of  a  curve.  In  Ex.  2  of  the  last 
article,  for  example,  the  new  equation  y^  =  3  x  is  simpler  than  the 
original  equation,  and  from  its  form  we  recognize  the  curve  as  a 
parabola.  It  is  obvious,  however,  that  the  position  of  the  new 
origin  is  of  fundamental  importance  in  thus  simplifying  the  equa- 
tion, and  we  shall  now  solve  an  example  illustrating  a  method  of 
determining  the  new  origin  to  advantage. 

Ex.  Transform  the  equation  y^  —  iy  -x^  —  Sx^  —  3x-|-3  =  0to  new  axes 
parallel  respectively  to  the  original  axes,  so  choosing  the  origin  that  there  shall 
be  no  terms  of  the  first  degree  in  x  and  y  in  the  new  equation. 


CHANGE  OF  ORIGIN  219 

The  formulas  of  transformation  are 

X  =  jco  +  x'    and    y  =  yo  +  y', 
where  suitable  values  of  Xq  and  yo  are  to  be  determined.    The  equation  becomes 

(2/0  +  VT  -  4  (yo  +  yl  -  {xo  +  a;')^  -  3  (xq  +  x'f  -  3  {xo  +  x')  +  3  =  0, 
or,  after  expanding  and  collecting  like  terms, 

y'^  +  (2  2/0  -  4)2/'  -  x'3  -  (3xo  +  3)x'2  -  (Sx^  +  6xo  +  3)x' 
+  (2/o  -  42/0  -  x^  -  3x2  _  3a.^  +  3)  =  0. 

By  the  conditions  of  the  problem  we  are  to  choose  Xo  and  yo  so  that    , 

2  yo  -  4  =  0,         3x2  +  6xo  +  3  =  0, 

two  equations  from  which  we  find  Xo  =  —  1  and  r/o  =  2. 

Therefore  (—1,  2)  should  be  chosen  as  the  new  origin  of  axes,  and  the  new 
equation  is  y'^  —  x'^  —  0,  or  y'^  —  x^,  after  the  primes  are  dropped. 

114.  In  particular,  this  method  of  simplifying  an  equation  is 
of  considerable  importance  in  studying  the  conies  defined  in 
Chap.  VII.    For  consider  the  equation 

a"       '^       V  '  ^  ' 

If  we  place  x  =  Xq  +  x',  y  =  y^  +  y',  (1)  becomes 

^  +  |f  =  1^  (2) 

a        0 

which  is  the  equation  of  an  ellipse  with  its  center  at  x'  =  0,  y'  =  0, 
and  its  axes  along  O'X'  and  O'Y'.    Therefore  (1)  is  an  ellipse' 
with  its  center  at  x  =  x^,  y  =  y^,  and  its   axes .  parallel  to   OX 
and  OY.  V^^ITft^ 

Furthermore,  if  a  >  &,  e  =  ■ ;  and  the  foci  of  the  ellipse 

a 

are  at  (x-'  =  ±  ae,  y'  =  0),  or,  what  is  the  same  thing,  (a;  =  ±  ae  +  x^, 
y  =  y^).    The  directrices  are  x'  =  ±  -  >  or  x  =  x^  ::r  -■ 
In  a  similar  manner 


220  CHANGE  OF  COORDINATE  AXES 

is  the  equation  of  an  hyperbola  with  its  center  at  {x^^  y^  and  its 
axes  parallel  to  OAT  and  OY;  and 

represents  a  parabola  with  its  vertex  at  {x^,  y^  and  its  axis  parallel 
to  OX. 

Any  equation  which  can  be  reduced  to  a  form  similar  to  one 
of  these  can  be  discussed  in  a  similar  manner.  A  general  treat- 
ment of  such  equations  will  be  found  in  Chap.  XI.  We  shall 
give  here  some  examples. 

Ex.1.    16x2 +  252/2  + 64a; -1502/ -111  =  0. 
Rewriting,  we  have 

16(a;2  +  4a;)+  25(2/2  _  6 y)  =  111, 
whence  16(x2  +  4x  +  4)  +  25(2/2  -  62/  +  9)  =  400, 

or  (x  +  2)2      (2/-3)2^^ 

25  16 

Placing  now  x  =  —  2  +  x',        y  —  Z  -\-  y', 

we  have  \-^—  =  \. 

25       16 

This  is  an  ellipse  with  semiaxes  5  and  4,  and  eccentricity  |.  Its  center  is  at 
(x'  =  0,  2/'  =  0),  its  foci  are  at  (x'  =  ±  3,  y'  =  0),  and  its  directrices  are  x'  =  ±  '^^- 

=  ±83- 

Hence  the  original  equation  represents  an  ellipse  with  semiaxes  5,  4,  and 

eccentricity  3.    Its  center  is  at  (—  2,  3),  its  foci  are  (—5,  3)  and  (1,  3),  and  its 

directrices  are  x  =  —  10^  and  x  =  6^. 

Ex.2.    5^2  _  102/ -  4x -7  =  0. 

Rewriting,  we  have  6  (2/2  —  2  ?/)  =  4  (x  +  |), 

5(2/2_2y  +  l)  =  4(x  +  |+|), 
or  (2/  -  1)2  =  4  (X  +  3). 

Placing  now  x  =  —  3  +  x', 

2/  =  1  +  2/', 
we  have  7/2  =  4  x'. 

which  represents  a  parabola  with  vertex  (x'=  0,  2/'=  0).    Its  axis  is  along  (YX'; 
its  focus  is  (x'  =  ^,  2/'  =  0),  and  its  directrix  is  x'  =  -  1. 

Therefore  the  original  equation  represents  a  parabola  with  its  vertex  at 
(—3,  1)  and  its  axis  parallel  to  OX.  Its  focus  is  (—24,  1)  and  its  directrix  is 
a;  =  -  3.L 


CHANGE  OF  DIRECTION  OF  AXES  221 

Ex.  3.     (X  -  C)2  +  2/2  -  e2a;-2. 

This  is  the  equation  of  tlie  conic,  as  found  in  §  81.    We  may  write  it  as 
(1  -  e2)a;2  _  2 ex  +  2/2  =  -  A 

Then  if  e ^i  1,  we  may  proceed  as  follows: 

(1  -  e2)  (x^  -  -l^x  + ]  +  2/2  ^  _  c2  +  _£!_  , 

(1  -  e2)  (x —Y+  2/2  =  -^ , 

^V        l-e2/^^       l-e2 


2/-* 
+  — ^ —  =  1. 
c2e2 


(1  -  e2)2  1  _  e2 

(<2g2 

We  may  now  place  =  a^, 

(1  -  e2)2 

=  a2(l-e2)  =  ±62^ 


1  -  e2 
and 


1  -  e2      e 
the  sign  of  6  being  ±  1  according  as  e  <  1.    The  equation  is  then 


a2  62 

The  equation  accordingly  represents  an  ellipse  or  an  hyperbola  with  center 
at  0,0). 

If  e  =  1,  the  equation            (x  —  c)2  +  y^  =  ^x^ 
becomes  y^  =  2cx  -  c*  =  2 c  (x 1 , 

which  represents  a  parabola  with  the  vertex  at  /- ,  0| . 

115.  Change  of  direction  of  axes  without  change  of  origin. 

Case  I.  Rotation  of  axes.  Let  OX  and  OY  (fig.  131)  be  the 
original  axes,  and  OX'  and  0  Y'  be  the  new  axes,  making  Z  </> 
with  OX  and  Oy  respectively.  Then  ZXOY'  =  90°  +  <f>,  and 
ZYOX' ^90° -(f>. 


222 


CHANGE  OF  COOKDl^ATE  AXE« 


Let  P  be  any  point  in  the  plane,  its  coordinates  being  x  and  y 
with  respect  to  OX  and  OY,  and  x'  and  y'  with  respect  to  OA"' 

and  0  F'.  Then  by  construction  OM  =  x, 
0N=  y,  OM'  =  x',  and  J/'P  =  y'.  Draw 
OP. 

The  projection  of  OP  on  OX  is  OM,  and 
the  projection  of  the  broken  line  OM'P 
on  OX  is  OM'  cos(f>  +  i»/'P  cos  (90°  +  <f>) 
-X  or  OM'  cos  ^  —  3f'Psin^. 

.-.  0Jf=01f' cos (^-J/'P sin (^,  (1) 

by  §  15. 
In  like  manner  the  projection  of  OF  on  OY  is  ON,  and  the 
projection  of  the  broken  line  OM'P  on  OF  is  Olf' cos  (90°  —  <^) 
+  J/'Pcos</). 

.'.  0N=  OM'  sin ^+M'P cos 0,  (2) 

by  §  15. 

Replacing  OM,  02i,  OM',  •  ••  by  their  values,  we  have 

X  =  x'  cos  (f>~  y'  sin  ^, 
y  =  a;'  sin^  +  y'  cos^. 

Ex.  1.    Transform  the  equation  xy  =  5  to  new  axes,  having  the  same  origin 
and  making  an  angle  of  45°  with  the  original  axes. 

x' 


Here  <p  =  45°,  and  the  formulas  of  transformation  are  x  = 


w'  x'  +  y' 


V2    '"  V2 

Substituting  and  simplifying,  we  have  as  the  new  equation  x?  —  y"^  =  10, 
from  which  we  recognize  the  curve  to  be  an  equilateral  hyperbola. 

Ex.  2.  Transform  the  equation  34x2  j^  \\yi  —  24x2/  =  100  to  new  axes  with 
the  same  origin,  so  choosing  the  angle  0  that  the  new  equation  shall  have  no 
term  in  xy. 

The  formulas  of  transformation  are 

X  =  x'  cos  <t>  —  y'  sin  0, 

y  =  x'  sin  0  +  y'  cos  0, 
where  0  is  to  be  determined. 

Substituting  in  the  equation  and  collecting  like  terms,  we  have 
(34  cos2  0  +  4 1  sin2  0  —  24  sin  0  cos  0)  x^ 

+  (34  sin2  0  +  41  cos2  0  +  24  sin  0  cos  0)  t/2 

+  (24  sin2  0  +  14  sin  0  cos  0  -  24  cos2  0)xj/  =  100. 

By  the  conditions  of  the  problem  we  are  to  choose  0  so  that 
24  sin2  0  +  14  sin  0  cos  0  -  24  cos2  0  =  0. 


OBLIQUE  COOKDINATES  223 

One  value  of  0  satisfying  this  equation  is  tan-i  |.  Accordingly  we  substitute 
sin^  =  I  and  cos^  =  |,  when  the  equation  reduces  to  x^  +  2y^  =  4,  which  is 
the  equation  of  an  ellipse. 

Case  II.  Interchange  of  axes.  If  the  axes  of  x  and  y  are  simply 
interchanged,  their  directions  are  changed,  and  hence  such  a  trans- 
formation is  of  the  type  imder  consideration  in  this  article.  The 
formulas  for  such  a  transformation  are  evidently  x  =  y',  y  =  x'. 

Case  III.  Rotation  and  interchange  of  axes.  Finally,  if  the 
axes  are  rotated  through  an  angle  <^  and  then  interchanged,  the 
formulas,  being  merely  a  combination  of  the  two  already  found,  are 

x  =  y'  cos<f>  —  x'  sin0,         y  =  y'  sin^  +  x'  cos^. 

A  special  case  of  some  importance  occurs  when  cf)=  270°.  We 
have  then  x  =  x',  y  =  —  y'. 

Cases  II  and  III,  it  should  be  added,  occur  much  less  frequently 
than  Case  I. 

In  case  both  the  origin  and  the  direction  of  the  axes  are  to  be 
changed,  the  processes  may  evidently  be  performed  successively, 
preferably  in  this  order:  (1)  change  of  origin;  (2)  change  of 
direction. 

116.  Oblique  coordinates.  Up  to  the  present  time  we  have 
always  constructed  the  coordinate  axes  at  right  angles  to  each 
other.  This  is  not  necessary,  however, 
and  in  some  problems,  indeed,  it  is  of 
advantage  to  make  the  axes  intersect 
at  some  other  angle.  Accordingly,  in 
fig.  132,  let  OX  and  OY  intersect  at 
some  angle  &>  other  than  90°. 

We  now  define  x  for  any  point  in  the 
plane  as  the  distance  from  OY  to  the 
point,  measured  parallel  to  OX;   and  y 

as  the  distance  from  OX  to  the  point,  measured  parallel  to  OY. 
The  algebraic  signs  are  determined  according  to  the  same  rules  as 
were  adopted  in  §  16. 

It  is  immediately  evident  that  the  rectangular  coordinates  are 
but  a  special  case  of  this  new  type  of  coordinates,  called  oUique 


224 


CHANGE  OF  COORDINATE  AXES 


coordinates,  since  the  new  definitions  of  x  and  y  include  those 
previously  given.  In  fact,  the  term  Cartesian  or  rectilinear  co- 
ordinates includes  both  the  rectangular  and  the  oblique. 

Oblique  coordinates  are  usually  less  convenient  than  the  rectan- 
gular, and  are  very  little  used  in  this  book.  If  necessary,  the 
formulas  obtained  by  using  rectangular  coordinates  can  be  trans- 
formed into  similar  ones  in  oblique  coordinates  by  the  formulas 
of  the  following  article.  When  no  angle  is  specified  the  angle 
between  the  axes  is  understood  to  be  a  risht  angle. 

117.  Change  from  rectangular  to  oblique  axes  without  change 
of  origin.    Let  OTand  OY  (fig.  133)  be  the  original  axes  at  right 

angles  to  each  other,  and 
OX^  and  OY'  the  new  axes, 
making  angles  <^  and  (/>' 
,  respectively  with  OX. 
n  Then  co  =  cf)'  —  ^.  Let  P 
be  any  point  in  the  plane, 
X  its  rectangular  coordinates 
being  x  and  y,  and  its  ob- 
lique coordinates  being  x' 
and  y'.  Draw  PM  parallel  to  OY,  PM'  parallel  to  OY',  M'N 
parallel  to  OY,  and  RM'N'  parallel  to  OX.    Then  Z  RM'P  =  <\>'. 

But  OM  =0N  +  NM  =0N+  M'N'  =  031'  cos  <b+3I'P  cos  (f>', 
MP  =  MN'  +  N'P  =  NM'  +  N'P  =  OM'  sm  <^  -F  M'P  sin  <^'. 

.*•  x=  x' cos<f>-{- y' cos(f>', 
y  =  x'  sin  <f>  +  y'  sin  (f>'. 


Ex.    Transfonn  the  hyperbola 


x-* 


a2      62 


=  1  to  its  asymptotes  as  axes. 


Since  the  equations  of  the  asymptotes  are  2/  =  ±  -  x,  0  =  tan- '  /  —  1 ,  and 
(f/  =  tan-i  -  ,  if  we  choose  to  have  the  hyperbola  lie  in  the  first  and  the  third 
quadrants  with  respect  to  the  new  axes.   The  formulas  of  transformation  become 

b 


Va-2  +  62 


{X'  +  y'),       y  = 


Va-  +  62 


i-x'  +  y'). 


Substituting  and  simplifying,  we  have  as  the  new  equation  xy 

Unless  b  —  a,  the  axes  are  obli(iue  and  w  =  2  taii^'  -  • 

a 


a2  +  62 


PROBLEMS  225 

118.  Degree  of  the  transformed  equation.  In  reviewing  this 
chapter  we  see  that  the  expressions  fur  the  original  coordinates  in 
terms  of  the  new  are  all  of  the  first  degree.  Hence  the  result  of 
an}'  transformation  cannot  be  of  higher  degree  than  that  of  the 
origmal  equation.  On  the  other  hand,  the  result  cannot  be  of 
lower  degree  than  that  of  the  original  equation ;  for  it  is  evident 
that  if  any  equation  is  transformed  to  new  axes  and  then  back  to 
the  original  axes,  it  must  resume  its  original  form  exactly.  Hence 
if  the  degree  had  been  lowered  by  the  first  transformation,  it  must 
be  increased  to  its  original  value  by  the  second  transformation. 
But  this  is  impossible,  as  we  have  just  noted. 

It  follows  that  the  degree  of  an  equation  is  unchanged  by  any 
single  transformation  of  coordinates,  or  by  any  number  of  succes- 
sive transformations.  In  particular,  the  proposition  that  any  equa- 
tion of  the  first  degree  represents  a  straight  line  is  true  for  oblique 
as  for  rectangular  coordinates. 

PROBLEMS 

1.  What  are  the  new  coordinates  of  the  points  (2,  3),  {—  4,  5),  and  (5,  —  7) 
if  the  origin  is  transferred  to  the  point  (.3,  —  2),  the  new  axes  being  parallel  to 
the  old  ? 

2.  Transform  the  equation  x^  +  Ay^  — 2x  +  8y  +  l  =  0to  new  axes  parallel 
to  the  old  axes  and  meeting  at  the  point  (1,  —  1)  with  respect  to  the  old  axes. 

3.  Transform  the  equation  ?/3  -  6?/2  +  3a;2  +  12  ?/ -  18x  +  35  =  0  to  new 
axes  parallel  to  the  original  axes  and  meeting  at  (2,  —  3)  with  respect  to  the 
original  axes. 

4.  Find  the  equation  of  the  ellipse  when  the  origin  is  taken  at  the  lower 
extremity  of  the  minor  axis,  and  the  minor  axis  is  the  axis  of  y. 

5.  Find  the  equation  of  the  ellipse  when  the  origin  is  at  the  left-hand  vertex, 
the  major  axis  lying  along  OX. 

6.  Find  the  equation  of  the  hyperbola  when  the  origin  is  at  the  left-hand 
vertex,  the  transverse  axis  lying  along  OX. 

7.  Find  the  equation  of  the  strophoid  when  the  origin  is  at  A  (fig.  92),  the 
axes  being  parallel  to  those  of  §  84. 

8.  Find  the  equation  of  the  strophoid  when  the  asymptote  is  the  axis  of  y, 
the  axis  of  x  being  as  in  §  84. 

9.  Find  the  equation  of  the  witch  (fig.  90)  when  LK  is  the  axis  of  x  and 
OA  the  axis  of  y. 


226  CHANGE  OF  COORDINATE  AXES 

10.  Find  the  equation  of  tlie  witch  when  the  origin  is  taken  at  the  center  of 
the  circle  used  in  constructing  it,  the  axes  being  parallel  to  those  of  §  82. 

11.  Find  the  equation  of  the  cissoid  when  its  asymptote  is  the  axis  of  y  and 
its  axis  is  the  axis  of  x. 

12.  Find  the  equation  of  the  cissoid  when  the  origin  is  at  the  center  of  the 
circle  used  in  its  definition,  the  direction  of  the  axes  being  as  in  §  83. 

13.  Find  the  equation  of  the  parabola  when  the  origin  is  at  t'.io  fo  -as  and 
the  axis  of  x  is  the  axis  of  the  curve. 

14.  Find  the  equation  of  the  parabola  when  the  axis  of  the  curve  and  the 
directrix  are  taken  as  the  axes  of  x  and  y  respectively. 

15.  Transform  ?/2  —  8x  —  10  2/  +  1  =  0  to  new  axes  parallel  to  the  old,  so 
choosing  the  origin  that  the  new  equation  shall  contain  only  terms  in  y^  and  x. 

16.  Transform  the  equation  12  x^  +  18  y^  _  12  x  +  12  ?/  —  31  =  0  to  new  axes 
parallel  to  the  old,  so  choosing  the  origin  that  there  shall  be  no  terms  of  the  first 
degree  in  the  new  equation. 

17.  Show  that  any  equation  of  the  form  xy  +  ax  +  6j/  +  c  =  0  can  always 
be  reduced  to  the  form  xy  =  k  hj  choosing  new  axes  parallel  to  the  old,  and 
determine  the  value  of  A;. 

18.  Show  that  the  equation  ax^  +  by^  +  ex  +  dy  +  e  =  0  (a ?i 0,  hjtiQi)  can 
always  be  put  in  the  form  ax^  +  hy'^  =  fc  by  choosing  new  axes  parallel  to  the 
old,  and  determine  the  value  of  k. 

19.  Show  that  the  equation  y^  +  ay  +  bx  +  c  =  Q  (^  ?^  0)  can  always  be 
reduced  to  the  form  y^  +  6x  =  0  by  choosing  new  axes  parallel  to  the  given  ones. 

20.  Find  the  equation  of  an  ellipse  if  its  axes  are  6  and  2,  its  center  is  at 
(—3,  2),  and  its  major  axis  is  parallel  to  OX. 

21.  Find  the  equation  of  an  ellip.se  if  its  axes  are  a  and  ^,  its  center  is  at 
(—  2,  —  3),  and  its  major  axis  is  parallel  to  OX. 

22.  Find  the  equation  of  an  hyperbola  if  its  transverse  axis  is  4,  its  conju- 
gate axis  2,  its  center  at  (1,  —  2),  and  its  transverse  axis  parallel  to  OX. 

23.  Find  the  equation  of  an  hyperbola  if  its  transverse  axis  is  v2,  its  con- 
jugate axis  V  §,  its  center  at  (2,  3),  and  its  transverse  axis  parallel  to  OX. 

24.  The  vertex  of  a  parabola  is  at  (3,  —  2)  and  its  focus  is  at  (5,  —  2).  Find 
its  equation. 

25.  The  vertex  of  a  parabola  is  at  (4,  5)  and  its  focus  is  at  (4,  1).  Find  its 
equation. 

26.  The  center  of  an  ellipse  is  at  the  point  (2,  3),  its  eccentricity  is  ^,  and 
the  length  of  its  major  axis,  which  is  parallel  to  the  axis  of  x,  is  10.  What  is 
the  equation  of  the  ellipse  ? 

27.  Find  the  equation  of  an  ellipse  when  the  vertices  are  (—2,  0),  (4,  0),  and 
one  focus  is  at  the  origin. 


PROBLEMS  227 

28.  The  center  of  an  hyperbola  is  at  (-  1,  -  2),  its  eccentricity  is  1^,  and  its 
transverse  axis,  which  is  parallel  to  OX,  is  4.    Find  its  equation. 

29.  The  vertex  of  a  parabola  is  at  the  point  (-  4,  -  2),  and  it  passes  through 
the  origin  of  coordinates.    Find  its  equation,  its  axis  being  parallel  to  OX. 

30.  Given  the  ellipse  4x^  +  9y^  +  8x  -  S6y  +  i -0;  find  its  eccentricity, 
center,  vertices,  foci,  and  directrices. 

31.  Given  the  ellipse  3a;2  +  5y2  +  i8  x  -  20y  +  32  =  0  ;  find  its  eccentricity, 
center,  vertices,  foci,  and  directrices. 

32.  Given  the  hyperbola  9 x^  —  4 y^  —  6ix  —  32 y  —  19  =  0 ;  find  its  eccen- 
tricity, center,  vertices,  foci,  directrices,  and  asymptotes. 

33.  Given  the  hyperbola  3x2  —  2  2/2  ^  6x  +  8y— 11  =  0;  find  its  eccentricity, 
center,  vertices,  foci,  directrices,  and  asymptotes. 

34.  Given  the  parabola  72x2  +  48x  +  180y  -  37  =  0;  find  its  vertex,  focus, 
axis,  and  directrix. 

35.  Given  the  parabola  y^  —  5x  +  6y  —  1  —  0;  find  its  vertex,  focus,  axis, 
and  directrix. 

36.  What  are  the  coordinates  of  the  points  (0,  1),  (1,  0),  (1,  1)  if  the  axes 
are  rotated  through  an  angle  of  60°  ? 

37.  Transform  the  equation  3x2  +  3?/2  —  lOxy  +  8  =  0  to  a  new  set  of  axes 
by  rotating  the  original  axes  through  an  angle  of  45°,  the  origin  not  being 
changed. 

38.  Find  the  equation  of  the  folium  x^  +  y^  —  3  axy  =  0  after  the  axes  have 
been  rotated  through  an  angle  of  46°. 

39.  By  rotating  the  axes  through  an  angle  of  45°  and  changing  the  origin, 
prove  that  the  curve  x'  +  y^  =  «^  is  a  parabola. 

40.  Transform  6x2  —  12xy  +  10?/2  _  14  =  o  to  a  new  set  of  axes,  making 
an  angle  tan-i  |  with  the  origiual'set. 

41.  Show  that  the  equation  x2  +  ?/2  =  a^  will  be  unchanged  by  transforma^ 
tion  to  any  pair  of  rectangular  ^.xes,  if  the  origin  is  unchanged. 

42.  Transform  the  equation  x"^  -  y^  =  36  to  new  axes  bisecting  the  angles 
between  the  original  axes. 

43.  Transform  the  equation  ix'^  -  3xy  +  8y^  =  I  to  one  which  has  no 
xy-term,  by  rotating  the  axes  through  the  proper  angle. 

44.  By  rotating  the  axes  through  the  proper  angle  transform  the  equation 
3  x2  +  2  Vs  xy  +  ?/■-  +  2  X  -  2  V3  ?/  =  0  to  another  which  shall  have  no  term  in  xy. 

45.  Transform  the  equation 

x2  _  6  2/2  _  6  V3  xy  +  [2  +  12  V3]  X  +  [20  -  6  V3]  y  -  15  +  12  V3  =  0 

to  a  new  set  of  rectangular  axes  making  an  angle  of  60°  with  the  original  axes 
and  intersecting  at  the  point  (-1,  2)  with  respect  to  the  original  axes. 


228  CHANGE  OF  COORDINATE  AXES 

46.  Transform  the  equation  ix-  +  9y^  =  36  from  rectangular  axes  to  oblique 
axes  with  the  same  origin,  making  angles  tan-i^  and  tan-i(—  J)  respectively 
with  OX. 

47.  Find  the  equation  of  the  hyperbola  Sx^  —  4y^  =  12  referred  to  its  asymp- 
totes as  coordinate  axes. 

48.  Show  that  the  lines  y  =  ±x  intersect  the  strophoid  at  the  origin  only,  and 
find  the  equation  of  the  curve  referred  to  these  lines  as  axes. 

49.  Transform  the  equation  2x^  —  Sy^  =  Q  from  rectangular  axes  to  oblique 
axes  having  the  same  origin  and  making  the  angles  tan-'  ^  and  tan-'  ^  respec- 
tively with  OX. 

50.  Prove  that  the  formulas  for  transposing  from  a  set  of  rectangular  axes 
to  a  set  of  oblique  axes  having  the  same  origin  and  the  same  axis  of  x  are 

x  =  x'  +  y'  cos  w, 
y  =  y'  sin  w, 
where  w  is  the  angle  between  the  oblique  axes. 

51.  By  transforming  the  equation  y  =  mx  +  6  by  the  formulas  of  example 
60,  show  that  the  equation  of  a  straight  line  in  oblique  coordinates  is 

sind> 
sin(«  —  <f>) 

where  u  is  the  angle  between  OX  and  OY,  <f>  the  angle  between  the  line  and  OX, 
and  c  the  intercept  on  OY. 

52.  Derive  the  result  of  example  51  directly  by  use  of  the  trigonometric 
formulas  connecting  the  sides  and  the  angles  of  an  oblique  triangle. 

53.  By  use  of  the  transformation  of  example  60,  prove  that  the  equation  of 
a  circle  in  oblique  coordinates  is 

(X  -  d)2  +  (y  _  e)2  4-  2 (X  -  d)  (y  -  e)cos u  =  r^, 

where  w  is  the  angle  between  the  axes,  and  (d,  e)  is  the  center. 

54.  Obtain  the  result  of  example  53  directly  by  use  of  the  trigonometric 
relations  connecting  the  sides  and  the  angles -of  an  oblique  triangle. 


CHAPTEE  XI 
THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE 

119.  Introduction.  The  most  general  equation  of  the  second 
degree  is  of  the  form 

Ax'+  2  Hxi/  +  Bf-{-2Gx+2Fy+C=0, 

where  the  coefficients  may  have  any  values,  including  zero,  except 
that  A,  B,  and  H  cannot  be  zero  together. 

We  shall  proceed  to  show  that  this  equation  always  represents 
an  ellipse,  an  hyperbola,  a  parabola,  or  a  limiting  case  of  one  of 
these,  if  it  represents  any  curve,  and  shall  derive  criteria  by  which 
the  nature  of  tlie  curve  can  be  readily  determined. 

120.  Removal  of  the  xy-t&rm.  Let  us  make  a  transformation 
of  coordinates  to  new  rectangular  axes,  making  an  angle  ^  with 
the  original  ones,  the  origin  being  unchanged.  The  formulas  of 
transformation  are  (§  115) 

x  =  x'  cos  4>  —  l/'  sin  (f), 
y  =  x'  sincf)  +  y'  cos  (f>. 

Substituting,  we  have 

A'x'''  +  2  H'x'y'  +  B'y'^  +  2  G'x'  +  2  F'y'  +  C"  =  0, 

where  A'  =  A  cos^ ^  +  2H sin (f>  cos<f) -\-B  sin^ </>, 

H'  =  (B  —  A)  sin  (}>  cos  (f)-\-H  (cos^<^  —  sin^</)), 

J5' =  ^  sin*^  0  —  2  ^  sin  ^  cos  ^  + -B  cos^  0, 

(T'  =  G^cos<^  +  i^sint/); 

i^'  =  ii^ cos </>-(?  sine/), 

and  C'=C. 

229 


230   GENERAL  EQUATION  OF  SECOND  DEGREE 

We  may  now  determine  <f)  so  that  H'  shall  vanish;  that  is, 
so  that 

2{B  —  A)  cos  <^  sin  <^  +  2  H(cos'(f>  -  sm^cfy)  =  0. 

This  equation  is  equivalent  to 

2  ^  cos  2  <^  +  (5  -  ^)  sin  2  <^  =  0, 
2ff 


whence  tan  2(f>  = 


A-B 

2H 


or  ^  =  J  tan  ' 


A-B 

To  compute  the  values  of  A^  and  B\  we  have 

A!  =A  cos^<f>+  2Hsia.<l)  cos(f>  +B  sia^<f> 

.l+cos2<^           .                 1—  cos2<^ 
=  A -+B:sin2(t>+B 

=  ^[^+5  +  (^-i?)cos2<^4- 2^sin20]. 

2H 


But,  since  tan  2  ^  = 


A-B 


sin  2  <^  =  ±  "'  >     cos  2  <^  =  ± 


and  therefore     A!  ^Wa-^B±   (^-^f+4^1 
2L  V(^-J5f+4^'J 


=  1  [^  +  j5  ±  V(^-^f+4^']. 


Sunilarly,       5'  =  ^  [^  4-  -S  :f  ^  {A-Bf^- A.H''\ 

From  these  results  it  follows  that 

A'B'  =  AB-H^. 

Hence  if  AB  —  H^  is  positive,  A'  and  ^'  have  the  same  sign ;  if 
AB  —  H^  is  negative,  ^'  and  B'  have  opposite  signs ;  if  AB  —  ^^ 
is  zero,  either  A'  or  B'  is  zero. 


EQUATION   WITHOUT  THE  xy-TERM  231 

The  discussion  of  the  general  equation  is  then  reduced  to  that 
of  the  simpler  equation 

This  equation  we  will  consider  in  the  next  two  articles,  dropping 
the  primes  for  convenience. 

121.  The  equation  Ax""  +  By"  -\-  2  Gx  +  2  Fy-\-  C  =  0.  We 
shall  prove  the  theorem :  The  equation 

Aaf  +  Bif+2Gx+2Fy  +  C=0, 

where  the  coefficients  are  such  that 

AF^'  +  BG^-ABC^^  0, 

represents  a  conic,  if  it  represents  any  curve  at  all.    In  particular, 

(1)  when  A  and  B  have  the  same  sign,  it  represents  an  ellipse  * 
or  no  curve  ; 

(2)  when  A  and  B  have  opposite  signs,  it  represents  an  hyperhola  ; 

(3)  when  either  A  or  B  is  zero,  it  represents  a  parabola. 

Suppose  first  that  neither  ^  nor  5  is  zero.  Then  the  equation 
may  be  rearranged  as  follows : 

A(cc'+2~x\+B(f+2^y)  =  -C. 

We  may  then  complete  the  squares  of  the  expressions  in  the 
parentheses;  thus, 


Aj  Y      BJ  AB 

*  The  circle  is  considered  a  special  case  of  an  ellipse  (see  §  75). 


232   GENERAL  EQUATION  OF  SECOND  DEGREE 

Since  AF'^  +  BG^  —  ABC  is  not  zero,  we  may  divide  by  the  right- 
hand  member  of  the  equation,  obtaining 


M  N 

where,  for  convenience,  we  place 

AF'  +  BG'^  —  ABC 


M  = 


N  = 


A'B 

AF^  +  BG--ABC 
AB- 


We  may  now  transfer  the  origin  of  coordinates  to  the  point 

(C  F\ 
>  — -  b  the  new  axes  remaining  parallel  to  the  old,  by  the 

formulas  ^  ^ 

A         '         ^  B      ^ 

The  equation  is  then \-^  =  \. 

^  M      N 

Now  if  A  and  B  have  the  same  sign,  M  and  N  wUl  have  the 
same  sign.  If  this  sign  is  positive,  we  may  place  M=  a^,  N=  V^, 
and  the  equation  is 

d'   ^    h'  ' 

which  represents  an  ellipse. 

The  axes  of  the  ellipse  are  parallel  to  the  original  coordinate 

(C  ft'X 
J  — -  I  referred  to  the 

original  axes,    li  A=B,  the  ellipse  is  a  circle. 
If  M  and  N  are  both  negative,  the  equation 

^  +  ^'  =  1 
M       N 

can  be  satisfied  by  no  real  values  of  x  and  y. 


EQUATION  WITHOUT  THE  xy-TERM  233 

If  A  and  B  have  opposite  signs,  M  and  N  have  opposite  signs, 
and  we  may  place  either  M=  a^,  iV  =  —  h'\  or  J/  =  —  a~,  JV  =  V^, 

thus  obtaining  either  ,.,        ,., 

^-  _  /;  ^ 

a'       J)'         ' 

-^  +  1^  =  1' 

either  of  which  represents  an  hyperbola. 

The  axes  of  the  hyperbola  are  parallel  to  the  original  coordinate 

axes,  and  its  center  is  at  the  point  ( > ^  |  referred  to  the 

•     1  V     ^         ^/ 

origmai  axes.  ^  ^ 

The  first  and  the  second  parts  of  the  theorem  are  therefore  proved. 

Consider  now  the  case  in  wliich  either  tI  or  ^  is  zero.    If,  for 

example,  A  =  0,  B  ^  0,  the  equation  is 

Bf+2Gx-\-2Fij  +  C=0, 

and  the  condition  to  be  fulfilled  by  the  coefficients  is  BG^  ^  0, 
which  is  equivalent  to  G  "^  0,  since  B  cannot  be  zero. 
We  may  arrange  the  equation  as  follows : 

1^+2— y=— 2— X 

^         B^  B         B 

Completing  the  square,  we  have 

F\^         2GI     .     C  F^ 


y  +  -^)=--^{^  +  ^^- 


Bj  B\        2G      2  GB 

If  now  we  transform  to  a  new  origin  by  placing 

X— 1 \-  x',     y = — — + y> 

2G      2GB  ^         B      ^' 

we  have  ■y'^  = x', 

which  is  the  equation  of  a  parabola. 

Similarly,  if  ^  =  0  but  A^  <d,  the  equation  may  be  reduced  to 
the  form 


which  is  also  a  parabola. 


2F 

a?'2  =  -  —  y'. 


234   GENERAL  EQUATION  OF  SECOND  DEGREE 

In  each  case  the  axis  of  the  parabola  is  parallel  to  one  of  the 
original  coordinate  axes. 

Hence  the  third  part  of  the  theorem  is  proved. 

122.  The  limiting  cases.    AVe  shall  consider  now  the  equation 

Ax'  +  Bf+2Gx+2Fy  +  C=0 

when  the  coefficients  are  such  that 

AF''  +  BG''-ABC=0. 

The  figures  represented  are  limiting  cases  of  a  conic,  since  the 
equation  of  this  article  may  be  obtained  from  that  of  the  previous 
article  by  allowing  the  coefficients  to  change  in  such  a  way  that 
AF^  +  BG^—ABC  approaches  zero.    We  have  three  cases: 

1.  A  and  B  have  the  same  sign. 

By  proceeding  as  in  §  121,  we  may  put  the  equation  in  the  form 

and  if,  as  before,  we  place 

X  =  -—  +  X',  y  =  -  —  +  x', 

A  B 

we  have  Ax"^  +  By'""  =  0. 

Since  A  and  B  have  the  same  sign,  we  may  consider  them  as 
positive,  and  factor  the  equation  as  follows : 

(Vja:'  +  IVBiJ)  {y/lix'  —  iVBy')  =  0, 

which  is  satisfied  by  real  values  of  x'  and  y'  only  when  x'  =  0, 

G  F 

y  =  0,  or  in  the  old  coordinates  x  = >  ?/= ■• 

^  A    "^  B 

Hence  in  this  case  the  equation  represents  a  point.  This  may  be 
considered  the  limiting  case  of  the  ellipse. 

2.  A  and  B  have  opposite  signs.  We  may  put  the  equation  in 
the  form  ,        ^2        /        r^\2 

^(.  +  _')  +  5(,+  _)  =  0, 

or  Ax"'  +  By"'  =  0. 


THE  DETERMINANT  AB  -  H^  235 

Since  A  and  B  have  opposite  signs,  we  will  consider  A  as  posi- 
tive and  B  as  negative.  The  equation  can  then  be  separated  into 
two  real  factors 

{VAx'  +  y/'^y')  {VAx'  ~  V^y')  =  0. 

Consequently  the  equation  represents  the  two  straight  lines  inter- 

sectmg  in  the  point  ic'  =  0,  y'  =  0,  or  a?  = >  2/  = 

This  may  be  considered  the  limiting  case  of  the  hyperbola, 

3.  One  of  the  coefficients  ^  or  -B  is  zero.    For  example,  let 

^  =  0,  ^  ^  0.    Then  the  condition  AF''  +  BG^-ABC=0  becomes 

G  =  0.    Hence  the  equation  is 

Bf+2Fi/-{-e=0. 
This  may  be  factored  into 

B{y-y^{y-y^)  =  ^, 

and  accordingly  represents  either  two  parallel  straight  lines,  two 
coincident  straight  lines,  or  no  real  locus,  according  as  y^  and  y^ 
are  real  and  unequal,  real  and  equal,  or  imaginary. 

This  is  considered  a  limiting  case  of  the  parabola. 

123.  The  determinant  AB  —  //'.  Returning  now  to  the  gen- 
eral equation  of  the  second  degree, 

Ax^+2Hxy-{-Bf+2Gx+2Fy  +  C=0, 

and  remembering  that  if  it  is  reduced  to  the  form 

^ V  +  B'y""  +  2  G'x'  +  2  F'y'  +  C'  =  Oy 
we  have  AB-H^'^A'  B', 

we  may  state  the  foUo^snng  theorem : 

The  equation 

Ac(^+2JH'xy+By^+2Gx+2Fy-\-C=0 

always  represents  a  conic  or  one  of  the  limiting  cases,  if  it  repre- 
sents any  curve  at  all. 


286      GENERAL  EQUATION  OF  SECOND  DEGREE 

1.  If  AB—H'  >  0,  the  equation  represents  an  ellipse,  a  point, 
or  no  curiae. 

2.  If  AB  —  H^  <  0,  the  equation  represents  an  hyperbola  or  two 
intersecting  straight  lines. 

3.  If  AB  —H^  =  0,  the  equation  represents  a  parabola,  two  par- 
allel lines,  two  coincident  lines,  or  no  curve. 

124.  The  discriminant  of  the  general  equation.  We  have  seen 
in  §  122  that 

A'x'''  +  B'y'^  +  2  G'x'  +  2  F'y'  +  C  =  0  (1) 

represents  one  of  the  limiting  cases  of  the  conic  sections  when 

A'F'^  +  B'G''-A'B'C'  =  0. 

It  is  useful  to  have  this  condition  in  terms  of  the  coefhcients  of 
the  general  equation 

Ax'-^2Hxy  +  By''^-2Gx  +  2Fy-\-C=0.  (2) 

This  might  be  done  by  substituting  for  A',  B',  G' ,  F',  and  C  the 
values  given  in  §  120,  but  this  method  is  tedious.  We  may  obtam 
the  result  by  noticing  that  the  first  member  of  (1)  can  be  factored 
rationally  in  x  and  y  when  it  represents  a  limiting  case,  and  not 
otherwise.  The  same  must  be  true  of  equation  (2).  We  shall  pro- 
ceed then  to  find  the  condition  under  which  (2)  can  be  factored. 

1.  Assume  ^  ^  0.  (2)  may  now  be  considered  as  a  quadratic 
equation  in  x,  and  factored  by  the  method  of  §  41.  Solving  (2) 
for  X,  we  have 


^-{Hy+  G)  ±-^/JlP-AB)y'+  2  y{HG-AF)  +  {G'-CA) 


It  is  necessary,  however,  that  y  should  not  appear  under  the  radi- 
cal sign,  and  for  this  it  is  necessary  and  sufficient  that  the  quantity 
under  the  radical  sign  must  be  a  perfect  square.  The  necessary 
and  sufficient  condition  for  this  is  (§  37) 

{HG-AF)--{H''-AB){G''-CA)=Q, 
that  is,  ylBC+2FGH-AF^-BG^-Cff^  =  0.  (3) 


CLASSIFICATION  OF  CONICS  237 

2.  Assume  ^  =  0,  but  B  ^^  0.  The  equation  may  then  be  con- 
sidered as  a  quadratic  equation  in  y,  and  handled  in  the  same 
manner  as  before  with  the  same  result. 

3.  Assume  A  =  (),  B  =  0.  Then  If  cannot  equal  zero.  The 
equation  can  consequently  be  written 

r*         V  c 

The  factors  of  this,  if  they  exist  at  all,  are  clearly  of  the  form 
{x+a){y  +  h)^Q, 

whence  a  =  —  >        5  =  — »        ah  = 

H  H  2H 

The  necessary  and  sufficient  condition  that  two  quantities  a  and  h 
can  be  found  satisfying  these  equations  is 

But  this  is  just  what  (3)  becomes  when  ^  =  0,  ^  =  0.    Hence, 
the  necessary  and  sicfflcient  condition  that 

Ax^+2Hxy  +  By^-\-2Gx+2Fy  +  C=Q 

represents  a  limiting  case  of  a  conic  is 

ABC +2  FGH  -AF^-  B  G^  -  6'//'  =  0. 

The  expression  (3)  is  called  tlie  discriminant  of  (1)  and  is 
denoted  by  A.    In  determinant  form 


A  = 


125.  Classification  of  curves  of  the  second  degree.  The  results 
of  the  previous  articles  are  exhibited  in  the  table  on  the  following 
page,  which  gives  the  simplest  forms  to  which  the  general  equation 

Ax}+  2Hxy+Bf+  2Gx+2Fy  +  C=0 

can  be  reduced  under  the  various  hypotheses,  where 

D  =  AB-H\ 

A  =ABC+  2FGH-AF^-BG--CIP. 


A 

H    G 

H 

B     F 

G 

F     C 

238      GENERAL  EQUATION  OF  SECOND  DEGREE 


At^O 

A  =  0 

D>0 

l2        7/2 

or  no  curve 

a;2      7/2 

D<0 

x2         ?/2 

Hyperbola- -^  =  1, 

X2         w2 
-^2  +  ^  =  1 

Two  Intersecting  straight  lines 

02        62 

D  =  0 

Parabola  2/2  =  4  pa;, 
or                 x^  =  4py 

Two  parallel  straight  lines 

{y  -  2/i)  {y  -  2/2)  =  0, 
or               (x  -  a;i)  (x  -  X2)  =  0, 
or  no  locus 

126.  Center  of  a  conic.  It  is  frequently  desirable  to  find  the 
center  of  a  conic  represented  by  the  general  equation.  Now,  if 
the  origin  of  coordinates  is  taken  at  the  center  of  the  curve,  the 
equation  can  contain  no  terras  of  the  first  degree  in  x  and  y ;  for 
if  it  is  satisfied  by  any  point  (x^,  y^,  it  must  also  be  satisfied  by 
the  symmetrically  placed  point  (—  x^,  —  y^.  We  will  accordingly 
take  the  center  £is  (ic^,  y^  and  make  the  transformation 

The  general  equation  then  becomes 

Ax'^  +  2  Hx'y'  +  By"  +  2  {Ax,  +  Hy,  +  G)x'+2  {Hx,  +  By,  +  F)  y' 
+  ^<+  2Hx^,  +  Byl+  2  Gx,+  2Fy,+  C=0, 

where,  by  the  condition  for  the  center, 

Ax,  +  ffy,+  G  =  0, 

irx,-{-By,  +  F=0. 

By  multiplying  each  of  these  by  a  properly  chosen  factor  and  add 
ing,  we  obtain  the  equivalent  equations 


(1) 


(AB  -  H"")  x^  =  HF-  BG, 
(AB  -  H')  y^=HG-  AF. 


(2) 


CENTEK  OF  A  CONIC  239 

Three  cases  then  occur: 

1.  AB  —  H^'^O.  Equations  (2)  have  then  a  single  solution 
and  the  curve  has  a  center.  This  occurs  for  the  ellipse,  the 
hyperbola,  and  their  limiting  cases. 

2.  AB  —  IT^  =  0,  but  not  each  of  the  expressions  HF—BG 
and  HG—AF  equal  to  zero.  At  least  one  of  equations  (2)  ex- 
presses an  absurdity,  and  hence  equations  (1)  have  no  solution 
and  the  curve  has  no  center.  This  occurs  in  the  case  of  the 
parabola. 

3.  AB-IT''  =  0,  JTF-BG=0,  ITG-AF=0.  Equations  (2) 
are  each  0  =  0.  Equations  (1)  are  identical,  and  any  point  on  the 
line  expressed  by  each  of  them  is  a  center  of  the  curve.  In  this 
case  one  easily  calculates  that  A  =  0.  The  curve  then  consists  of 
two  parallel  straight  lines  (§  125),  and  the  line  of  centers  is  the 
line  halfway  between  the  two  parallel  lines. 

127.  If  for  the  equation 

Ax'+2  Hxy  +  Bf+1Gx+2Fy  +  C=0 

the  origin  is  transferred  to  the  center  of  the  curve,  when  such 
exists,  the  equation  becomes 

Ax'^+2Hx'y'  +  By'''+C' =  Q, 
where         C  =  Ax^  +  2  Hx,y,  +  Byl  +2Gx^+2  Fy^  +  C. 

This  quantity  C  may  be  expressed  in  terms  of  the  original  coeffi- 
•  cients  as  follows.    Take  the  equations  (1)  of  §  126,  multiply  the 
first  one  by  x^,  the  second  by  y^,  and  add  them.    There  results 

Ax^  +  2  Hx^,  +  By^  +  Gx,  +  Fy,  =  (i. 

Subtracting  this  from  the  value  of  C',  as  given  above,  we  have 

C'  =  Gx,  +  Fy,+  C, 

whence,  by  substituting  the  values  of  x^  and  y^,  as  given  by  (2) 
(§  126),  we  have 

ABC+2FGH-AF^-BG^-Cff^      A 


C'  = 


AB-H'  D 


240   GENERAL  EQUATION  OF  SECOND  DEGREE 

128.  Directions  for  handling  numerical  equations.  In  case  it 
is  necessary  to  reduce  a  uumerical  equation  to  its  simplest  form, 
the  procedure,  based  on  the  foregoing  discussion,  is  as  follows : 

First  compute  AB  —  H^  and  determine  the  type  of  the  curve 
(§  123).    A  may  also  be  computed  if  wished,  but  it  is  not  necessary. 

If  AB  —  H^  ^  0,  find  the  center,  as  in  §  126,  and  transfer  the 
origin  to  it.    Then,  as  in  §  120,  turn  the  axes  through  an  angle 

(p  =  w  tan     =  tan 


A-B  ■s/(^A-Bf+4:H''±{A-B) 

computing  A'  and  B'  by  the  formulas  of  §  120.     The  two  values 
of  tan^  are  the  slopes  of  the  axes  of  the  curve. 
If  AB  —  H'^  =  0,  write  the  equation  in  the  form 

{y/~Ax  +  ^yf+2Gx  +  2Fy  +  C=Q, 
Vb  being  taken  with  the  same  sign  as  H,  and  let 

,      -y/Ax  +  V^y  ,      Vbx  —  VAv 

y'  =  — 7=^'       «'  =  — , ^- 

■■Va  +  b  ^a  +  b 

Solve  these  equations  for  x  and  y  and  substitute  in  the  given 
equation. 

The  equation  is  now  in  the  form  y'^-i-  2  G'x'  +  2F'y'  +  C"  =  0, 
and  the  further  reduction  is  made  by  the  method  of  §  121. 

Ex.1.    8x2  _  4xy  +  52/'2-36x  + 18?/ +  9  =  0. 

Here  AB  —  11^  =  30,  and  the  curve  is  an  ellipse  or  a  limiting  case  of  an  ellipse. 
The  center,  found  by  §  126,  is  (2,  —  1),  and  the  equation  transferred  to  the 
center  as  origin  becomes 

8  x'2  -  4  xY  +  5  ?/'2  -  36  =  0. 

We  now  turn  the  axes  through  (p  =  ^  tan-i(—  |)  =  tan-i2  or  tan-i(—  ^), 
and  find,  from  §  120,  ^'  =  9  or  4,  B'  =4  or  9. 

The  ambiguity  is  removed  by  noticing  that  if  we  take  tan  (f>  —  2,  the  formulas 
of  transformation  (§  115)  are 

,      z"  —  2  ij"  ,      2  x"  +  y" 
2/  = 


which  give  A'  =  4,  B'  =  9. 
The  simplest  equation  is  then 

4  x"2  +  9  ?/"2  =  36. 
The  slopes  of  the  axes  are  2  and  —  J^, 


CONIC  THROUGH  FIVE  POINTS 


241 


Ex.  2.    36a;2-48x2/  +  Wy^  +  52x  -  260y  -  39  =  0. 

(6  X  -  4  r/)2  4-  52  a;  -  260  y  -  39  =  0, 
6x  —  4?/      Sx  —  2y 


Here 
We  write 

and  place 


y 


V52  Vl3 

4x  —  Gy       —  2x 


8y 


V52  Vl3 

Solving  for  x  and  y  and  substituting,  we  have 

y'S  4.  VlSj/'  +  Vl3x'  -3=0, 
or  y''2  =  —  VTs  x", 

Vl3 
^~ 

The  curve  is  a  parabola,  the  axis  of  which  is  y"  =  0  or  6  x 


where 


Vl3 


y  = 


¥ 


4  2/  +  13  =  0. 


129.  Equation  of  a  conic  through  five  points.    The   general 
equation  of  the  second  degree 

1x^+2  Hxy  +  By^+2Gx  +  2Fy  +  C=Q 

contains  six  constants,  the  ratios  of  which  are  alone  essential.  Five 
independent  equations  are  sufficient  to  determine  these  ratios. 
Therefore  a  conic  is,  in  general,  determined  by  five  conditions.  The 
simplest  conditions  are  that  the  conic  should  pass  through  the  five 
points  (a?„  y^,  [x^,  y^,  (x^,  y^),  {x^,  y^,  and  {x.^,  y,).  The  five  equa- 
tions to  determine  the  ratios  of  A,  If,  B,  G,  F,  and  C  are  then 

Axl  +  2  Hx^y^  +  Byf  +  2Gx^+ 2Fy^+ C  =  0, 
Axl+2Hx^y^  +  By^+  2  Gx^+2Fy^+C  =  0, 
Axl+  2Hx,y^  +  By^+  2  Gx^+2Fy,+  C  ={), 
Axl+2Hx^^  +  Byl+  2  Gx^+  2Fy,  +  C=0, 
Ax^  +  2 Hx^y^  +  Byl -^  2Gx^+ 2Fy^+C  =  Q. 

Eliminating  the  coefficients  between  these  and  the  general  equa- 
tion, we  have 


=  0, 


x-" 

xy 

f 

X 

y 

xl 

^iVx 

yl 

^1 

2/1 

^! 

^^y^ 

yl 

x^ 

y. 

^^8 

^^z 

yl 

x^ 

2/3 

x! 

^43/4 

yl 

X, 

y^ 

< 

^6^5 

yl 

X, 

y. 

242   GENERAL  EQUATION  OF  SECOND  DEGREE 

which  is  the  required  equation  of  a  conic  through  five  given  points. 

The  equation  of  a  conic  through  five  points  may  also  be  found 
in  the  following  manner : 

Let  us  take  any  four  of  the  given  points  and  connect  them  by 
straight  lines  so  as  to  form  a  quadrilateral  (fig.  134). 

Let  the  equation  of  I^J^  be  A^x  +  B^y  +  C^  =  0,  or,  more  shortly, 
f^{x,  y)  —  0.  Similarly,  let  the  equation  of  P^l^  be  fj^x,  y)  =  0,  that 
of  ^^  be  f.J^x,  y)  =  0,  and  that  of  F^J^  be  f^{x,  y)  =  0. 

Form  now  the  equation 

Vi{x,  y)  ■  Ai-^,  y)  +  ¥2(«.  y)  ■  A{^>  y)  =  o.  (i) 

where  I  and  k  are  undetermined  factors.    This  equation  is  of  the 

second  degree  in  x  and  y ;  therefore 
it  represents  a  conic  section.  More- 
over, this  conic  section  passes  through 
Il\  for  the  coordinates  of  P^  make 
f^{x,  y)  =  0  and  f^{x,  y)  =  0,  and 
therefore  satisfy  equation  (1).  Simi- 
larly, this  conic  passes  through  ^, 
^,  and  j^.  If  now  we  substitute  in 
(I)  the  coordinates  of  j^,  we  de- 
termine values  of  I  and  k,  which  we 
must  assume  in  order  that  the  conic 
may  pass  through  i^.  We  thus  de- 
termine the  equation  of  a  conic  through  the  five  given  points. 

Ex.  Let  it  be  required  to  pass  a  conic  through  the  points  Pi(2,  3),  P^i  —  1,  2), 
P3(-3,  -1),  P4(0,  -4),  P5(l,  1). 

The  equation  of  P1P2  isa;-3y  +  7  =  0,  that  of  P2P3  is3x-22/  +  7=:0, 
that  of  PsP4  is  X  +  y  +  i  =:  0,  and  that  of  P4P1  is7a;-2y-8  =  0. 

We  form  the  equation 

l{x  -Sy  +  '!)(x  +  y  +  4)  +  k(Sx  -  2y +  'J){7  x  -  2y  -  8)  =  0, 

and,  substituting  the  coordinates  of  P5,  find  k  =  ^l. 
Hence  the  required  conic  is 

(X  -  3  2/  +  7)  (a;  +  2/  +  4)  +  I  (3  a;  -  2  y  +  7)  (7  X  -  2  y  -  8)  =  0, 
or  109x2-108x2/ +  82/2  + 169x -10?/ -168  =  0. 

If  three  of  the  points  lie  in  a  straight  line,  the  method  is  appli- 
cable, but  it  is  evident  that  the  conic  must  be  one  of  the  limiting 


CONIC  THKOUGH  FIVE  POINTS  243 

cases,  for  it  must  consist  of  the  straight  line  in  which  the  three 
points  lie,  and  the  straight  line  connecting  the  other  two  points. 

If  four  or  five  of  the  points  lie  in  a  straight  line,  the  method  is 
not  applicable.  It  is  geometrically  evident  that  in  this  case  the 
problem  is  indeterminate  ;  for  the  conic  may  consist  of  the  straight 
line  in  which  the  four  points  lie,  together  with  any  line  through  the 
fifth  point,  if  that  is  not  on  the  line  with  the  four,  or  any  line  what- 
ever if  the  fifth  point  lies  on  a  straight  line  with  the  four  others. 

If  it  is  required  to  determine  a  parabola,  only  four  points  are 
necessary.  This  follows  from  the  fact  that  one  relation  connecting 
the  coefficients  is  always  given,  namely,  AB  —  H^  =  0.  We  form, 
as  before,  the  equation 

^/i(«'.  y)  ■  M^>  y)  +  ^M^>  y)  ■  f^i-^^  y)  =  o- 

We  form,  then,  the  equation  AB  —  H'^  =  0  out  of  the  coefficients 
of  this  equation.    The  result  is  a  quadratic  equation  in  ->  and 

K 

hence  we  will  have  two,  one,  or  no  real  parabolas,  according  as  the 
values  of  j  are  real,  equal,  or  imaginary.    It  should  be  noticed  that 

fC 

in  this  connection  "  parabola  "  may  mean  two  parallel  straight  lines. 

Ex.  Let  it  be  required  to  pass  a  parabola  tlirough  the  points  Pi(l,  —  1), 
P2(2,  3),  Ps(2,  -  6),  P4(5,  7). 

We  find  the  equations  of  the  following  lines :  P1P2,  4x  —  y  —  6  =  0;  P2P3, 
X  -  2  =  0 ;  P3P4,  4x-2/-13  =  0;  P4P1,  2a;-y-3  =  0.  The  equation  of 
the  conic  is  then 

Z(4x  -  2/  -  5)  (4x  -  2/  -  13)  +  fc(a;  -  2)(2x  -  2/  -  3)  =  0, 
or  (16Z  +  2A:)x2  +  (-8Z- A:)x2/  +  i2/2 

+  ( _  72  J  -  7  fc)  X  +  (18  i  +  2  i)  2/  4-  «5  Z  +  6  fc  =  0, 

and  the  condition  AB  —  R"^  =  d'xs 

(16Z  +  2A:)Z-(4Z  +  ^A:)2  =  0, 
whence  k-Oov—%1. 

There  are  accordingly  two  parabolas, 

16x2  -  8 X2/  +  2/2  -  72 X  +  18 2/  +  65  =  0, 
and  2/=^-16x  +  2  2/  +  17  =  0. 

The  first  equation,  however,  represents  a  limiting  case  of  a  parabola,  since 
it  factors  into  4x_2/-5  =  0    and    4x-2/-13  =  0, 

which  represent  two  parallel  straight  lines. 


244      GENEEAL  EQUATION  OF  SECOND  DEGEEE 

130.  Oblique  coordinates.  We  have  assumed,  thus  far,  that  the 
general  equation  is  referred  to  rectangular  coordinates.  If,  how- 
ever, the  equation 

Ax"  +  2  Hxy  +  Bf+2Gx-j-2Fy  +  C=0 

has  reference  to  oblique  coordinates,  it  may  be  transformed  to  any 
conveniently  chosen  pair  of  rectangular  coordinates.  Formulas  for 
this  purpose  are  given  in  §  117,  and  it  has  been  proved  in  §  118 
that  such  a  transformation  does  not  alter  the  degree  of  the  equa- 
tion.   Therefore  the  new  equation  is  of  the  form 

A'x'^  +  2  H'x'y'  4-  B'y""  +  2  G'x'  +  2  F'y'  +  C'=0. 

This  equation  may  now  be  investigated  by  the  methods  of  this 
chapter. 

Hence  we  have  the  result : 

Any  equation  of  the  second  degree,  whether  referred  to  rectangu- 
lar or  to  oblique  coordinates,  represents  a  conic. 

PROBLEMS 

Determine  the  nature  and  the  position  of  the  following  conies  : 

1.  4xj/  +  .3  2/2_8x  +  16?/ +  19  =  0. 

2.  a;2  _  6xy  +  9  2/2 -280«- 20  =  0. 

3.  Ilx2_4xy  +  14i/2_26x  +  32y  +  59  =  0. 

4.  5x2  -  26xy  +  51/2  +  10x-2Gy  +  71  =  0. 

5.  4:xy  +  6x-8y  +  1  =  0. 

6.  x2-2x2/  + y2  +  2x-22/  +  l  =  0. 

7.  13x2  +  10 xy  -}- 13  2/2  +  6x  -  42y  -  27  =  0. 

8.  x2 -4xy  -  2  2/2- 14x  + 4  2/  + 25  =  0. 

9.  6x2  -  5x2/ -  62/2 -46x- 9?/ +  60  =  0. 

10.  4  x2  -  8  x?/  +  4  2/2  +  6  X  -  8  2/  +  1  =  0. 

11.  x2  +  6x2/ +  92/2 -6x -18  2/ +  5  =  0. 

12.  41x2  -  24x2/  +  342/2  -  188x  +  116?/  +  196  =  0. 

13.  31  x2  -  24  X2/  +  21 2/2  +  48x  -  84 2/  +  84  =  0. 

14.  Show  that,  if  A  and  B  in  the  general  equation  have  opposite  signs,  the 
conic  is  an  hyperbola. 

15.  Show  that  the  conic  represented  by  the  general  equation  is  an  equilateral 
hyperbola  when  A  =  ~  B. 


PEOBLEMS  245 

16.  Prove  that  the  necessary  and  sufficient  conditions  that  the  general 
equation  should  represent  a  circle  are  A  =  B,  H  —  0,  provided  the  axes  are 
rectangular. 

17.  Show  that,  if  the  general  equation  contains  the  term  in  xy  and  not  more 
than  one  of  the  terms  containing  x^  or  y^,  the  conic  is  an  hyperbola. 

18.  Show  that  xy  +  ax  +  by  +  c  =  0  is  the  general  equation  of  the  hyperbola 
when  the  axes  of  coordinates  are  parallel  to  the  asymptotes. ,_.     ^  ...  - 

19.  Prove  that  any  homogeneous  equation  in  x  and  y  represents  a  system  of 
straight  lines  passing  through  the  origin. 

20.  Find  the  angle  between  the  two  straight  lines  represented  by  the  equation 
Ax^  +  2  Hxy  +  By^  =  0. 

21.  Show  that  the  asymptotes  of  the  hyperbola  are  parallel  to  the  two 
straight  lines  Ax^  +  2  Hxy  +  By^  =  0, 

22.  Show  that,  if  the  focus  is  taken  upon  the  directrix,  the  conic  becomes 
one  of  the  limiting  cases. 

Find  the  equations  of  the  conies  through  the  following  points : 

23.  (3,  2),  (-  2,  -  3),  (J,  -  3),  (2,  -  2),  (|,  -  f). 

24.  (1,  2),  (6,  3),  (3,  2),  (2,  1),  (9,  2). 

25.  (0,  a),  (a,  0),  (0,  -«),{-  a,  0),  (a,  a). 

26.  (1,  1),  (-1,6),  (2,4),  (0,3),  (3,1). 

27.  Find  the  equation  of  a  parabola  through  the  four  points  (4,  —  4),  (9,  4), 
(6,  -  1),  (5,  -  2). 

28.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from  two 
intersecting  straight  lines  is  constant.  Prove  that  the  locus  is  an  ellipse,  and  find 
its  eccentricity  in  terms  of  the  angle  between  the  lines. 


CHAPTER  XII 

TANGENT,  POLAR,  AND  DIAMETER  FOR  CURVES  OF  THE 
SECOND  DEGREE 

131.  Equation  of  a  tangent.    It  has  been  shown  in  §  59  that 
the  tangent  to  a  curve  at  a  point  {x^,  y^  is 

where  ( —  )  denotes  the  value  of  -^  at  (x.,  yX 
\dx/i  ax 

Applying  this  theorem  to  the  conic 

Aa?+2  Hxy  +  Bf+ 2Gx+2Fy +  C  =  0, 
we  first  find,  by  differentiation, 

2Ax-h2Hy+2Hx^  +  2By^  +  2G+2F^=0, 
ax  ax  ax 

.  dy  Ax  +  Hy  +  G 

whence  ,    =  —  -77 tt 7; ' 

dx  Hx+By  +  F 

Therefore  the  equation  of  the  tangent  at  the  point  {x^,  y^  is 

Ax^  +  Hy^+G.  , 

y  —  y,  = ^^ ix  —  X.), 

^      ^'  Bx^  +  By^  +  F^  ''' 

that  is,        Ax^x  —  Ax^  +  Hxy^  +  Hx^y  —  2  Hx^y^  +  ^ViV  ~  ^Vl 
+  Gx  —  Gx^  +  Fy  —  Fy^  =  0. 

This  equation  may  be  simplified  by  adding  to  it  the  identity 

Ax*-^  2Hx^y^  +  Byl+  2Gx^+2Fy^  +  C=  0, 

which  follows  from  the  fact  that  {x^,  y^  is  on  the  conic.  There  results 

Ax^x  +  H{x^y  +  xy;)  +  By^y  +  G{x  + x^)  +  F{y  +  y;)+ C  =Q. 

This  result  is  easily  remembered  from  its  resemblance  to  the  equa- 
tion of  the  conic. 

246 


POLAR  247 

132.  Definition  and  equation  of  a  polar.    We  have  just  seen  in 
§  131  that  the  equation 

Ax^x  +  H(x,y  +  xy^)  +  By^y  +G{x+x;)  +  F(y  +  y,)  +  C  =  0     (1) 

represents  the  tangent  line  to  the  conic 

As(^-\-2Hxy+By^+2Gx  +  2Fy  +  C=0,  (2) 

provided  the  point  (x^,  y^  is  on  the  conic.  But  no  matter  what  is 
the  position  of  the  point  {x^,  y^,  (1),  being  of  the  first  degree,  repre- 
sents some  straight  line  which  from  the  form  of  the  equation  must 
in  some  way  be  related  to  the  conic  (2)  and  the  point  {x^,  y^. 

This  line  is  called  the  polar  of  the  point  {x^,  y^  with  respect  to 
the  conic,  and  the  point  is  called  the  pole  of  the  line. 

The  tangent  line  now  appears  as  only  the  special  case  of  the 
polar  which  occurs  when  the  pole  is  on  the  conic. 

Ex.  1.    The  polar  of  the  point  (3,  —  2)  with  respect  to  the  ellipse 
4a;2  +  5  2/2_2x  +  3y-l  =  0 
is  12x-10y-(x  +  3)  +  §(2/-2)-l  =  0, 

or  22x-17y-14  =  0. 

Ex.  2.    Find  the  pole  of  the  line  2  x  —  3^  +  6  =  0  with  respect  to  the  hyper- 
bola 4x2-5T/2  +  4x-2i/  +  3  =  0. 
The  polar  of  (Xj,  yi)  is 

4xix  -  5yiy  +  2{x  +  Xi)  -  (2/  +  2/i)  +  3  =  0, 
or  (4xi  +  2)x  +  (-  5yi  -l)y  +  2xi  -  yi  +  S  =  0. 

This  will  be  the  same  as  the  given  line  if 

4xi  +  2  ^  5?/i  +  1  _  2xi  -  yi  +  3 
2  3  6  ' 

These  reduce  to  the  two  equations  for  Xi  and  yi, 
12x1-102/1  +  4  =  0, 
2x1-11^1  +  1=0; 
whence  •      Xi  =  -  ^|,        ?/i  =  ^V- 

133.  Fundamental  theorem  on  polars.    When  the  equation 

Ax'+2Hxy+By^+2Gx+2Fy  +  C  =  0  (1) 

represents  one  of  the  limiting  cases  of  the  conies,  the  polar  has 
little  importance.    We  shall  therefore  assume  that  the  conic  is 


248 


TANGENT,  POLAE,  AND  DIAMETER 


either  an  ellipse  (including  the  circle),  a  parabola,  or  an  hyperbola. 
The  properties  of  its  poles  and  polars  are  then  conveniently  found 
by  use  of  the  proposition : 

If  P^is  any  'point  on  the  polar  of  another  point  ij,  the  polar  of 
II  passes  through  Py 

For  the  polar  of  Il(x^,  y^  with  respect  to  (1)  is 
Ax^x  +  H{x^y  +  xy^)  +  By^y  +G{x  +  x^)  +  F{y  +  y^)  +  C  =  0,  (2) 
and  if  P^ix^,  y^  is  on  (2),  we  must  have 
Ax^x^  +  II{x^y^-\-  x^y^  +  By^y^_  +  G  (^,+  x^  +  F{y^-^  y,)  +  C=  0.  (3) 

Again,  the  polar  of  P^  with  respect  to  (1)  is 

Ax^x  +  H{x^y  +  xy^)  +  By^y  j^G{x  +  x^}  +  F{y  +  7/,)  +  C  =  0,  (4) 

and  this  passes  through  {x^,  y^)  because  of  (3). 

134.  Chord  of  contact.  An  inspection  of  the  figures  of  the  conies 
shows  that  a  point  not  on  a  conic  must  lie  so  that  in  general  either 
two  tangents  or  no  tangent  can  be  drawn  from  it  to  the  conic.  In 
the  former  case  the  point  is  said  to  be  outside  the  conic ;  in  the 

latter  case,  inside.  Let  us  take 
now  a  point  7^  outside  the  conic, 
and  let  the  two  tangents  drawn 
from  it  to  the  conic  touch  the 
conic  in  L  and  K  (fig,  135).  Now 
the  polar  of  a  point  on  a  conic 
is  the  tangent  to  the  conic  at 
that  point  (§  132).  Hence  I^L  is 
the  polar  of  L,  and  I^K  is  the 
polar  of  K.  Therefore,  by  the  fundamental  theorem  (§  133),  the 
polar  of  7J  must  pass  through  L  and  K.  Hence  the  polar  is 
the  straight  line  LK,  which  is  called  the  chord  of  contact  of 
tangents  from  7J. 

Conversely,  if  a  straight  line  intersects  a  conic,  its  pole  is  the 
point  of  intersection  of  the  tangents  at  the  points  of  intersection. 
The  proof  of  this  is  left  to  the  student. 


POLAR 


249 


The  chord  of  contact  may  be  used  to  find  the  equations  of  the 
tangents  through  a  point  not  on  the  conic. 

Ex.  Find  the  tangents  to  the  conic  x^  +  2xy  +  y^  +  2x  +  6y  +  l  =  0  which 
pass  tlirougli  the  point  (4,  —  2). 

Since  this  point  is  not  on  the  conic,  its  coordinates  not  satisfying  the  equa- 
tion of  the  conic,  we  form  the  equation  of  its  polar,  i.e.  3x  +  5?/  —  1  =  0,  which 
will  be  the  chord  of  contact  of  the  tangents  drawn  from  the  point  to  the  conic, 
provided  any  can  be  drawn.  Solving  the  equations  of  the  polar  and  the  conic 
simultaneously,  we  find  that  they  intersect  at  the  points  (7,  —  4)  and  (2,  —  1). 

Hence  there  are  two  tangents  which  are  respectively  2x  +  3y  —  2  —  0  and 
x  +  2y  =  0.  •■■■'■     ■  ■•-   ■■■    -■•■■'•':  -   ,  ■ 

135.  CottstiUction  of  a.  ^lar.  Whethfer  a  point  lies  inside  or 
outside  a  conic,  the  polar  may  be  obtained  by  the  following  con- 
struction. Draw  through 
i^  (fig.  136)  two  straight 
lines,  one  intersecting  the 
conic  in  L  and  K,  and 

the  other  intersecting  the    /  /   ^y    ^^^^s 

conic  in  M  and  N.    Let 
the  tangents  at  L  and  K 
intersect  in  H  and   the 
tangents  at  M  and  iV  in- 
tersect in  S.    Then  B  is  the  pole  of  LK  and  S  is  the  pole  of  UN, 
by  §  134    Since  i^  lies  on  both  LK  and  3IN,  its  polar  passes 
through  B  and  S  by  the  fundamental  theorem.    Therefore  ES  is 
the  required  polar. 

This  construction  may  also  be  used  when  ^  is  outside  the  conic. 

136.  The  harmonic  property  of  polars.  An  important  property 
of  poles  and  polars  is  stated  in  the  theorem :  Any  secant  passing 

through  I^  is  divided  har- 
monically hy  the  conic  and 
the  polar  of  P^. 

Let -^iNT  (fig.  137)  be  any 
secant  through  P^,  M  and 
N  be  the  points  in  which 
P^N  cuts  the  conic,  and  Q 
the  point  in  which  it  cuts 
the  polar  of  P^.    We  are  to  prove  that  the  line  MN  is  divided 


Fig.  137 


250  TANGENT,  POLAR,  AND  DIAMETER 

harmonically,  i.e.  that  it  is  divided  externally  and  internally  in 
the  same  ratio.    We  are  to  prove,  then,  that 

P^M  _  MQ 
F^N~  Qjsr' 

whence,  by  placing  MQ  =F^Q—P^M,  QN=P^N—P^Q,  and  solving 
for  P^Q,  we  have  2 PM ■  PN 

^^       P^M+P^N 

Let  the  point  P^  be  {x^,  y^,  the  equation  of  the  conic  be 

Aa?-\-2Hxy  +  Bf+2Gx  +  2Fy  +  C=Q,  (1) 

and  that  of  the  polar  of  P^  be 
Ax^x  +  H{x^y  +  xy^  +  By^y  +  G{x  +  x;)  +  F{y  +  ^/j)  +  C  =  0.  (2) 

Let  {x,  y)  be  a  variable  point  on  F^N,  r  the  variable  distance  JJP, 
and  d  the  angle  made  by  F^N  and  OX.    Then 

.     ^-«^i  .    ^     y-Vx 

cos  V  = ,         sm  a  = , 

r  r 

that  is,  x  =  r  cos  0  +  x^,       y  =  r  sin.  0  ■{■  y^  (3) 

Now  if  P  coincides  with  either  M  or  iV,  the  values  of  x  and  y 
given  by  (3)  satisfy  (1).    Substitution  gives 

r^  [A  cos'  0+2II  sin  0cos0+B  sin'  0] 

+  2r  [Ax^  cos  0  +  -^'(iCj  sin  0  +  y^  cos  ^) 
+  %,  sin  ^  +  G!  cos  ^  +  i^  sin  6']  +  C"  =  0, 
where  C  =  Ax'-  +  2  Hx,y,  +  J5yf  +  2  (?a;i  +  2  Fy^  +  C. 

The  roots  of  this  equation  are  P^M  and  P^N.    Hence,  by  §  43, 

2  [J«,cos  ^  +//(«!  sin  ^  +  yj  cos0)-i-By^  sin  ^  +  (?cos^  ^-i^sin  ^] 


F^M-F^N 

whence 


A  cos' (9  +2  7/ sin  0cos0+B  sin'^ 


^  cos'^  +  2^sin  ^  cos  ^  +  5  sin^^ 

2F,M-F,N 

FJf+P,N  , 

(4) 

^a;jCos^+-H'(ajiSin^  +  2/jCOS^)+%iSin^  +  <9cos^+i^sin^    ^ 


EECIPROCAL  POLAES  251 

Also,  if  the  point  P  coincides  with  Q,  the  values  of  x  and  y 
given  by  (3)  satisfy  (2).    Substitution  gives 

r  [Ax^  cos  6  +  H{x^  smd  +  y^  cos  d)+By^  siD.d  +  Gco80  +  Fsin.  6] 
+  C"  =  0. 


The  root  of  this  is  F^Q.    Therefore  F^Q 

C 


(5) 


Ax^  cos  6-\-H(x^  sin  6  +  y^  cos  6) + By^  mid+G  cos  6  +i>''sin  6 

Comparing  (4)  and  (5),  we  have 

_  2F,MF^N 

^^~  f^m+f^n' 

which  was  to  be  proved. 

The  theorem  of  this  article  is  often  made  the  basis  of  the 
definition  of  the   polar. 

137.  Reciprocal  polars.  Consider  a  given  conic  and  a  rectilinear 
figure,  such  as  the  triangle  ABC  with  sides  a,  b,  c  (fig.  138).  Con- 
struct the  lines  a',  b',  c',  the  polars  of  A,  B,  C,  respectively  with 
respect  to  the  conic.  The  lines  a',  b',  c'  form  a  new  triangle  A'B'  C. 
The  fundamental  theorem  shows  that  A',  B',  C'  are  the  poles  of 


a,  b,  c  respectively.  Hence  the  two  triangles  are  so  related  that  the 
vertices  of  one  are  the  poles  of  the  sides  of  the  other.  They  are 
called  reciprocal  polars.  A  similar  construction  holds  for  any 
figure  composed  of  straight  lines. 

Consider  next  any  curve  K  and  a  tangent  line  a  (fig.  139).    Let 
A  be  the  pole  of  a  with  respect  to  a  conic  C.    As  the  tangent  rolls 


252 


TANGENT,  POLAR,  AND  DIAMETER 


around  the  curve  K,  the  point  A  describes  another  curve  h  Let 
a  and  h  be  two  tangents  to  K,  and  M  their  point  of  intersection, 
and  let  A  and  B  be  the  two  corresponding  points  of  k,  and  m  the 
chord  AB.    Then,  by  the  fundamental  theorem,  m  is  the  polar  of  M. 

Now  let  a  and  h  approach 
coincidence.    Then  M  ap- 
proaches a  point  on  K,  B 
and    A    approach    coinci- 
dence, and  m  approaclies  a 
tangent  to  h.    Hence  tlie 
points  of  K  are  the  poles 
of  the  tangents  to  k. 
We  have  then  two  curves  such  that  the  points  of  either  are  the 
poles  of  the  tangents  of  the  other.    These  curves  are  called  reciprocal 
polars. 

The  study  of  reciprocal  polars  forms  an  important  part  of  geom- 
etry, but  lies  outside  the  limits  of  this  work. 

138.  Definition  and  equation  of  a  diameter.    A  diameter  of  a 
conic  is  the  locus  of  the  iniddle  points  of  a  system  of  parallel  chords. 

I,     i^t 

Asg'+2  Hxy+  Bf-^  2  Gx 
+  2Fy  +  C=Q  (1) 

be  any  conic  (fig.  140),  RS  an^^ 
chord  which  makes  the  angle  6 
with  OX,  and  -?J(«i,  y-^  the  mid- 
dle point  of  this  chord.  Take 
P{x,y)  any  point  on  the  chord, 
and  let  P^P  =  r,  where  r  is  posi- 
tive if  iJP  has  the  direction  of 

as,  and  negative  if  I^P  has  the  direction  SB.  Then  for  any 
position  of  P  we  have 


-X 


Fig.  140 


=  cos  6, 

r  ' 


2/ -2^1 


sin^: 


whence 


X  =  x^+  r  cos  d,       y  =  y^+  r  sin  0. 


(2) 


DIAMETER  253 

Now  if  P  coincides  with  either  B  or  S,  the  values  of  x  and  y  in 
(2)  satisfy  (1).    Substituting,  we  have 

r"  [^  cos' (9  +  2  iT  sin  6^  cos  6"  +  ^  sm='^] 

+  2  r  [Ax^  cos  6  +  Hx^  sin  ^  +  Hy^  cos  6 

-\-By^&ind  +  Gco^d  +  F&md] 

+  [Axl  +  2  Hx^y^  +  %,2  ^  2  Gx^  +  2  i^'^/i  +  C]  =  0,     (3) 

the  roots  of  which  are  P^S  and  P^R.  But,  by  hypothesis,  P^R 
=  —  -^*S^.  Hence  the  roots  of  equation  (3)  are  equal  in  magnitude 
and  opposite  in  sign.  Therefore  the  coefficient  of  r  in  (3)  must  be 
zero,  that  is, 

Ax^  cos  6  +Hx^  sin  6  +  Hy^  cos  ^  +  %j  sin  6*  +  G^  cos  ^ + i<"  sin  ^  =  0.  (4) 

If,  for  convenience,  we  assume  that  cos  6^0,  and  this  will  gen- 
erally be  the  case,  we  may  divide  by  cos  0  and  replace  tan  0  by  the 
usual  symbol  for  the  slope  m,  thus  obtaining 

Ax^  +  Hy^  +G  +  m  {Hx^  +  By^  +  F)  =  0.  (5) 

If  we  allow  RS  to  move  parallel  to  itself,  so  that  m  remains 
fixed  but  P^  changes,  (5)  always  holds  true,  and  in  fact  shows  that 
7J  is  always  a  point  of  the  straight  line 

Ax  +  Hy  +  G  +  m{Hx  +  By+F)=Q.  (6) 

Conversely,  any  point  P^{Xy,  y^)  on  line  (6)  makes  the  values  of 
r  in  (3)  equal  in  magnitude  but  opposite  in  sign,  and  if  i^  lies  so 
that  these  roots  are  real,  it  will  be  the  middle  point  of  a  chord 
with  slope  m.  ,-.  g  lo  i?i.^s;r>  iii" 

The  straight  Ime  (6)  is  of  infinite  length,  and  it  is  customary  to 
regard  the  entire  line  as  the  diameter,  though  it  is  evident  that 
not  all  of  its  points  correspond  to  chords  of  the  system  which 
intersect  the  conic  in  real  points.  i^w  .-iuj  ui.na 

139.  The  last  statement  of  the  previous  article  may  be  explained  as  follows : 

The  equation  y  =  ??ix  +  6  may  be  made  to  represent  any  line  of  slope  m  by 

assigning  an  appropriate  value  to  b.    For  some  values  of  b  the  corresponding 

line  intersects  the  conic  (1)  of  §  138  in  real  points,  and  is  one  of  the  chords 

bisected  by  the  diameter  (6). 


254  TANGENT,  POLAK,  AND  DIAMETER 

For  other  values  of  6,  however,  the  line  does  not  intersect  the  conic  in  real 
points,  the  simultaneous  values  of  x  and  y  satisfying  their  equations  being 
imaginary.    But  if  these  imaginary  values  of  x  and   y  are  substituted  for 

Xi,   X2  and  yi,   yi  respectively   in   the   formulas  x  =  — ^ ,  y  =  ^ — —  of 

§  18,  the  resulting  values  of  x  and  y  are  real,  and  furthermore  they  satisfy  the 
equation  of  the  diameter. 

This  fact  is  sometimes  expressed  by  saying  that  the  line  is  a  choi'd  of  the 
conic  which  intersects  it  in  imaginary  points,  and  that  its  middle  point  is  a 
real  point  of  the  diameter.  It  is  from  this  point  of  view  that  the  entire  line  is 
regarded  as  the  diameter,  since  every  point  of  it  is  the  middle  point  of  some 
chord  of  the  system. 

140.  If  the  conic  has  a  center,  every  diameter  passes  through  the 
center.    For,  by  §  126,  the  center  satisfies  the  equations 

Ax  +  Hi/  +  G  =  0,         Hx+By  +  F=0, 

and  hence  satisfies  (6)  of  §  138  for  any  and  all  values  of  m. 

In  the  parabola,  hovjever,  all  diameters  arc  parallel  to  each  other 
and  to  the  axis  ;  for  the  slope  of  the  diameter  is,  from  (6),  §  138, 

A.  +  Hm  I 

But  for  the  parabola  H  =  ^ AB,  so  that  the  slope  of 

^  +  ^^  A      -J~A~  \l~ 

the  diameter  becomes --^= »  which  reduces  to ;=  • 

SAB  +  Bm  y/B 

This  is  independent  of  m,  and  equal  to  the  slope  of  the  axis  (§  128). 

It  is  evident  that  the  axes  of  a  conic  are  diameters,  for  from  the 
symmetry  of  the  curves  they  contain  the  middle  points  of  all 
chords  which  are  perpendicular  to  them.  In  fact,  they  are  the 
only  diameters  which  are  perpendicular  to  the  chords  which  they 
bisect,  as  will  be  proved  later  on. 

141.  Diameter  of  a  parabola.  If  the  equation  of  the  parabola 
is  written  in  its   simplest  form,  'if  =  4:px,  the  equation  of  the 

diameter  becomes  y  =  —  • 
m 

From  this  equation  it  is  evident  that  the  only  diameter  perpen- 
dicular to  the  chords  which  it  bisects  is  the  axis  of  the  parabola. 

Ex.  1.  Find  the  equation  of  the  diameter  of  the  parabola  2y2^,3x  =  0 
bisecting  chords  with  slope  2. 

Since  m  =  2  and  p  =  —  ^,  the  equation  of  the  diameter  is,  y  = ^  »  or 

2  2/  +  3  =  0.  ^ 


DIAMETER  OF  A  PARABOLA 


255 


Ex.  2.    A  diameter  of  the  parabola  y^  =  2x  passes  through  the  point  (2,  —  1). 

What  is  its  equation,  and  what  is  the  slope  of  the  chords  bisected  by  it  ? 

If  m  is  the  slope  of  the  chords  bisected,  the  equation  of  the  diameter  is 

y  —  —.    But  (2,  —  1)  is  a  point  of  this  diameter. 

m  1  J 

.-.  —  1  =  — ,  whence  rn  =  —  1 ;  also  the  diameter  is  y  = ,  ov  y  =  —  \. 

tn  —  1 

This  equation  of  the  diameter  could  have  been  written  down  immediately, 
for  the  diameter  is  parallel  to  OX,  so  that  if  one  of  its  points  is  distant  —  1  from 
OX,  all  its  points  are  distant  —  1  from  OX,  and  its  equation  isy  =  —  1. 

If  we  solve  the  equations   of  the  diameter  and  the  parabola 
simultaneously,  we  find  the  coordinates  of  0'  (fig.  141),  their  point 
p     2p^ 


of  intersection,  to  be 


The  equation  of  the  tangent  at  0'  is  found  to  be  3/  =  mx  + 


whence  it  is  seen  that  its  slope  is  m. 

Calling  0'  the  end  of  the  diameter,  we  express  the  above  theo- 
rem as  follows :  The  tangent  at  the  end  of  a  diameter  is  parallel  to 
the  chords  bisected  hy  the  diameter. 

If  we  consider  the  tangent  as  the  limiting  position  of  a  chord 
which  is  moved,  yet  retains  its  original  slope,  the  above  theorem 
seems  almost  immediately  evident. 

142.  Parabola  referred  to  a  diameter  and  a  tangent  as  axes.  Let 
O'X'  (fig.  141)  be  a  diameter  of  parabola 

f  =  ^px,  (1) 

bisecting  chords  of  slope  m,  and  O'Y'  be  the  tangent  at  0'.    Then 

p     1p^ 
^m^     m 
and  the  slope  of  O'Y'  is  m. 

First  transposing  (1)  to  O'X'  and 
O'Y",  where  O'Y"  is  parallel  to  OY, 
we  have  the  formulas  of  transfor- 
mation 


the  coordinates  of  0'  are 


x  =  ^,  +  x". 


m 


y  =  ^  +  y". 


The  new  form  of  the  equation  is 
y"^+^y"=4:px". 


m 


Fig,  141 


256  TANGENT,  POLAR,  AND  DIAMETER 

Using  now  the 'formulas  of  transformation  of  §  117,  which  become 

a?"  =x'+        ^        >         y"  =    /^^      > 

Vl  +  mt:    c:. . , '  -.:..:    .  .  V 1  +  tlV^ 

since  <^  =  0  and  <^'  =  tan" '  m,  we  have,  finally, 


,, . ; :  By  §  1 7,  however,        FO'  =  P(^-^/'^) . 

Therefore  if  we  denote  FO'  by  p',  after  dropping  the  primes  from 
«  and  2/,  the  equation  becomes 

y^  =  4  j?'x. 

^^  ""It  is  to  be  noted  that  an  equation  in  the  form  y"^  =  4:]px  always 
represents  a  parabola,  the  x  axis  being  a  diameter,  the  y  axis  a 
tangent,  and  the  distance  of  the  focus  from  the  origin  being  one 
fourth  the  coefficient  of  x. 

143.  Diameters  of  an  ellipse  and  an  hyperbola.    If  the  equation 

+  -.  ~  2  2 

of  the  ellipse  is  written  in  its  simplest  form,  —  +  ^  =  1,  and  the 

a^      If 

common  slope  of  the  chords  is  denoted  by  m^  the  equation  of  the 

diameter  becomes 

V 
V  = 7, —  X. 

If  the  slope  of  the  diameter  is  denoted  by  m,,  m^= — > 

whence  m,w„  = 5  • 

' .  -If  b  =^  a^  m^ni^  csLnnot  in  general  be  —1,  cmd  the  diameter  of  an 
ellipse  cannot  in  general  he  perpendictdar  to  the  chords  which  it 
bisects.  The  single  exception  is  when  the  chords  are  parallel  to 
either  axis,  in  which  case  the  diameter  is  the  other  axis  and  is 
perpendicular  to  the  chords  which  it  bisects,  as  noted  above. 

If  Z>  =  a,  the  ellipse  becomes  a  circle,  and  m^ni^  is  always  equal 
to  —  1.  Hence  the  diameter  of  a  circle  is  always  perpendicidar  to 
the  chords  which  it  bisects. 


DIAMETERS  OF  CENTRAL  CONICS 


257 


62 


jtti  being 
4 


becomes  — 
a^mi  9  mi 


Ex.  1.  Find  the  equation  of  a  diameter  of  the  ellipse  ix^  +  9y^  =  3(j  bisoft- 
ing  chords  parallel  to  the  line  a;-|-2y  +  l  =  0. 

Here   a^  =  9,   h"^  =  4,   and  in\  =  —  ^.    .-.  the  diameter  is  y  =  — 
or  9  2/-  8x  =  0. 

Ex.  2.    2y  +  3x  =  0isa  diameter  of  the  ellipse  4x2  +  9 ^2  =  3^ 
the  slope  of  the  chords  which  it  bisects  ? 

The  slope  of  the  diameter  is  —  |,  and  by  the  formula  is  — 

the  slope  of  the  chords  bisected.    As  a^  =  9  and  62  =  4,  — 

3  4,  8 

.-. = ,  whence  mi  =  — 

2  9  77H  27 

Ex.3.  Find  the  diameter  of  the  circle  4x2 +  4 2/2 +  43;  — By  — 11  =  0 
bisecting  chords  of  slope  2. 

The  center  of  the  circle  is  (—  -^,  1),  so  that  the  required  diameter  will  be 
2/  -  1  =  -  |(x  +  I),  or  2 X  +  4 ?/  -  3  =  0. 

Ex.  4.  Find  the  diameter  of  the  circle  4x2  +  42/2  +  4x  —  By  —  11=0,  which 
passes  through  the  point  (2,  —  1). 

The  center  of  the  circle  is  (—  ^,  1),  and  the  straight  line  determined  by  the 
two  points  (2,  —  1)  and  (—  i,  1),  i.e.  4x  +  oy  —  3  =  0,  is  the  required  diameter. 


In  the  case  of  the  hyperbola  —,  —  —, 


1  it  is  to  be  noticed  that 


the  parallel  chords  may  be  drawn  in  two  ways.  They  may  join 
points  on  the  same  branch  of  the  hyperbola,  or  points  of  one 
branch  to  points  of  the 
other  branch,  as  repre- 
sented in  fig.  142. 

In  whichever  way 
the  chords  are  drawn,  if 
their  common  slope  is 
denoted  by  m^,  the  equa- 
tion of  the  diameter  is 


y  = 


Fig.  142 


This  equation  differs  from  that  for  the  diameter  of  the  ellipse 
only  in  the  sign  of  the  right-hand  member.  .^ 

If  W2  is  the  slope  of  the  diameter,  m^m^  =  —  >  and,  as  in  the 

case  of  the  ellipse,  a  diameter  of  an  hyperbola  cannot  be  perpendic- 
ular to  the  chords  it  bisects,  except  in  the  two  special  cases  of  the 
transverse  axis  and  the  conjugate  axis. 


258  TANGENT,  POLAR,  AND  DIAMETER 

144.  Conjugate  diameters.  In  §  143  we  have  seen  that  if  the 
slope  of  the  chords  of  the  ellipse  —  +  ^  =  1  is  denoted  by  m^,  and 
the  slope  of  the  diameter  is  denoted  by  m^, 


w„  =  — 


am. 


whence     m^m^  =  — 


(1) 


Similarly,  if  the  slope  of  the  chords  is  m^,  the  slope  of  the  diam- 

eter  bisecting  them  must  be  —  -^ —  >  which,  by  (1),  must  be  m,. 

am\ 

Hence  the  proposition:    If  m^  and  m^  are  the  slojjes  of  two 

diameters  of  an  ellipse,  and 

rthj)n„  = > 


then  each  diameter  bisects  all 
chords  parallel  to  the  other. 
Such  diameters  are  called 
conjugate  diameters.  As  the 
major  and  the  minor  axis 
each  bisects  chords  parallel 


Fig.  143 


to  the  other,  they  are  conjugate  diameters. 
It  follows  that : 

1.  The  two  axes  are  the  only  pair  of  conjugate  diameters  which 
are  perpendicular  to  each  other. 

2.  If  one  of  two  conjugate  diameters  of  an  ellipse  makes  an 

acute  angle  with  the  axis  of  x,  the  other  makes  an  obtuse  angle 

h^ 
with  the  axis  of  x.    For  if  m^  >  0,  m^<  0,  since  m^m^= r^- 

But  a  positive  slope  corresponds  to  an  acute  angle,  and  a  negative 
slope  to  an  obtuse  angle.  Hence  the  upper  portions  of  conjugate 
diameters  always  lie  on  opposite  sides  of  the  minor  axis,  as  OA^ 
and  OB^  in  fig.  143,  A^A^  and  B^B^  being  conjugate  diameters. 


In  similar  manner  for  the  hyperbola  — 
of  two  diameters  m^  and  m„  are  such  that 

m,m„  =  —  > 


y  _ 


=  l,if  the  slopes 


CONJUGATE  DIAMETERS 


259 


the  corresponding  diameters   are  conjugate,  and  each  bisects  all 
chords  parallel  to  the  other.    The  transverse  and  the  conjugate 
axes  are  conjugate  diameters,  each  of  which  bisects  chords  parallel 
to  the  other. 
It  follows  that : 

1.  The  two  axes  are  the  only  pair  of  conjugate  diameters  that 
are  perpendicular  to  each  other. 

2.  Two  conjugate  diameters  make  either  both  acute  or  both 
obtuse  angles  with  the  transverse  axis ;  for  m^m^  being  always 
positive,  vi^  and  m^  have  the  same  sign. 

3.  Two  conjugate  diameters  lie  on  opposite  sides  of  either  asymp- 

tote  ;  for  since  m.m„  =  —  >  if  ?»,  <  - >  then  m^>  - >  and  the  corre- 
^    ^      a^  a  'a 

sponding  conjugate  diameters  are  on  opposite  sides  of  the  asymptote 

2/  =  ^x(fig.l46). 

145.  Ellipse  and  hyperbola  referred  to  conjugate  diameters  as 
axes.    Let  the  conjugate  diameters  OA^  and  OB^  of  the  ellipse 


^  +  1  =  1 
a'^  b' 


(1) 


(fig.  144)  be  chosen  as  new 
axes  OX'  and  OY',  and  let 
them  make  angles  (f>  and  ^' 
respectively  with  OX. 

Then  the  formulas  of  trans- 
formation are 


x  =  x'  cos  <f>  +  y'  cos  <f>', 
y  =  x'  sin  4>  +  y'  sin  ^', 


(2) 


where 


Le. 


tan  <^  tan  <^ 


I  _ 


Fig.  144 
b^ 


sin  <^  sin  </>'      cos  <^  cos  <f>'  _  n 
b'         +  a'  "' 


(3) 


since  OX'  and  OY'  are  conjugate  diameters. 


260 


TANGENT,  POLAR,  AND  DIAMETER 


Substituting  in  (1)  and  collecting  like  terms,  we  have 

cos^(f>      sin^^X   ,2  ,  o/cos  ^  cos  (f>'      sin  (f>  sin  <f>'\  ,  , 


v 


+  ,'2^'  +  !«^V=l. 


x'y 


(4) 


But  the  coefficient  of  x'y'  is  zero,  by  virtue  of  (3) ;  and  if  the 
intercepts  on  OX'  and  OY'  are  denoted  by  a'  and  b'  respectively, 
i.e.  OA^  =  a'  and  OB^  =  b',  (4)  becomes 


4-  =^  =  1 


(5) 


where   a'  = 


and     &'  = 


Icos^ 


(f)      sin^<f) 


^l 


cos-(f)'       sm^<f)' 


The  equation  of  an  ellipse  can  assume  the  form  (5)  only  when 
the  axes  chosen  are  a  pair  of  conjugate  diameters,  as  only  then  will 
the  coefficient  of  xy  be  zero.  Conversely,  any  equation  in  form  (5) 
is  an  elhpse  referred  to  a  pair  of  conjugate  diameters  as  axes. 

In  similar  manner,  the  equation  of  the  hyperbola  referred  to  a 

pair  of  conjugate  diameters  as  axes  is  — ^  —  ^  =  1,  where  at  pres- 
ent no  geometrical  meaning  will  be  assigned  to  b'. 
146.  Properties  of  conjugate  diameters. 

1.  The  tangent  at  the  end  of  a  diameter  is  parallel  to  the  conjugate  diameter. 
We  shall  prove  the  theorem  for  the  ellipse,  the  same  form  of  proof  being  appli- 
cable to  the  hyperbola. 

In  fig.  145  let  Ai  have 
coordinates  (a;i,2/i).  Then 

the  slope  of  OAi  is  — , 
and  if  the  slope  of  OBx  is 
7/i2 ,  m2  —  = SI  whence 

Xi 


a2' 


b^Xi 
a^yi' 


m^  = s —    The  equa- 

tion  of  the  tangent  .4  iTi 

a.  ^,   «f  +  ?g^  =  l, 

the    slope    of    which    is 
Hence  this  tangent  is  parallel  to  the  conjugate  diameter  0B\. 


CONJUGATE  DIAMETERS  261 

2.  The  sum  of  the  squares  of  the  halves  of  two  conjugate  diameters  of  an  ellipse 

is  constant  and  equal  to  the  sum  of  the  squares  of  the  halves  of  the  major  and  the 

minor  axes,  i.e.  a"^  +  b"^  =  a^  +  h"^. 

b^Xi 
We  have  just  seen  that  the  slope  of  OBi  (fig.  146)  is ,  so  that  its 

equation  is  i,2r 

y  =  -^x.  (1) 

Solving  this  equation  simultaneously  with  the  equation  of  the  ellipse,  in  order 
to  find  coordinates  of  Bi,  we  substitute  the  value  of  y  from  (1)  in h  —  =  1- 

Ct  V 

As  a  result  x^  = — - — s-    But  Ai(xi,  yi)  is  a  point  of  the  ellipse,  so  that 

2  2  ""2/1  +  ^^^1 

^  +  ?^  =  1,  or  62x2  +  a2y  2  =  a2ft2. 

«'       ^'  2  =  ^  =  ^' 

•  •    *  a2^2  62     ' 

and  X  =±  —— ' 

b 

bxi 
By  substitution  in  (1),  2/  =  T 

Therefore  the  coordinates  of  J5i  are  ( -^ ,  — ^ )  • 

\        b       a  / 

If,  as  in  §  145,  we  denote  OAi  by  a'  and  OBi  by  b',  by  §  17, 
a'2  =  a;  2  +  7/2^ 

a2y2      62x2 
and  ^"  =  -^+^' 

and  hence  a'2  +  6'2  =  ^^1+^  x  ^  +  ^^^±^  y f 

a2  o2 

=  a^  +  f>^, 

X  V 

since  —  +  —  =  1,  as  noted  above, 
a2       62 

3.  The  area  of  the  parallelogram  formed  by  drawing  tangents  to  an  ellipse  at 
the  ends  of  conjugate  diameters  is  constant  and  equal  to  4  ab.  Let  Ti  T^  T-s  T4 
(fig.  145)  be  a  parallelogram  formed  by  the  tangents  at  the  ends  of  the  con- 
jugate diameters  A1A2  and  B1B2.  Now  the  area  of  this  parallelogram  is  evi- 
dently four  times  the  area  of  the  parallelogram  ^lOlJiTi.    But^iTi=  Oi'i 

\a*v^  4-  6*x^  iCiX 

=  6'  = ~ i  ,  from  work  above  ;  and  since  the  equation  of  AiTi  is  — - 

a^  22 

,   ViV  _  1  the  perpendicular  distance  from  0  to  ^  1  Ti  is,  by  §  32,  — ; -_  • 

62  "^    /  VaV  +  6V 

/^a*y^  +  ¥x^\/         a^b-^         \        ,         ,  ,, 
Hence  the  area  of  AiOBi Tj  =  ( \ ^-    I      ,  =  a&,  and  the 

\  a&  /\^a*yl  +  b*xl/ 

area  of  the  large  parallelogram  is  4  ab,  as  was  to  be  proved. 


262 


TANGENT,  POLAR,  AND  DIAMETER 


147.    It  was  noted  in  §  144  that  conjugate  diameters  of   the   hyperbola 

=  1  lie  on  opposite  sides  of  the  asymptotes,  whence  it  follows  that  if 

one  of  two  conjugate  diameters  intersects  the  hyperbola,  the  other  cannot  inter- 
sect it.  In  order,  then,  to  state  for  the  hyperbola  propositions  analogous  to  2  and 
3  of  the  last  article,  it  is  customary  to  consider,  in  connection  with  the  above 

hyperbola,  the  hyperbola 1 =  1.    These  two  hyperbolas  are  called  con- 

jugate  hyperbolas,  either  one  being  considered  the  primary  and  the  other  being 

called  the  conjugate. 

It  may  readily  be  proved  that  if  the  slopes  of  two  diameters  are  such  that 

62 
rmm^  =  — ,  they  are  conjugate  diameters  of  both  the  above  hyperbolas.    More- 

over  it  is  evident  (fig.  146) 
that  if  one  diameter  in- 
tersects one  hyperbola,  the 
other  intersects  the  conju- 
gate hyperbola. 

Now  if  OAi  and  0B\  are 
conjugate  diameters,  and 
OAx  is  called  o',  as  in 
§  145,  and  we  apply  the 
same  method  as  was  ap- 
plied to  the  ellipse,  we  shall 
find  OBi  =  6'  of  §  145. 

With    this   value    of   6', 
theorem  2,  §  146,  becomes 

for  the  hyperbola  a'2  —  6'2  =  a2  —  62,  while  theorem  3  is  the  same  for  the 

hyperbola  as  for  the  ellipse. 

The  proofs  of  these  last  statements  are  left  to  the  student,  the  work  being 

exactly  like  that  for  the  ellipse. 


Fig.  146 


PROBLEMS 

Find  the  polars  of  each  of  the  following  points  with  respect  to  the  given 
conic,  and  find  the  points  in  which  the  polar  intersects  the  conic : 

1.  (1,  2),  23x2  -\\xy  +  iy^  +  36x  -  9y  +  9  =  0. 

2.  (-1,  -2),  3x2-3xy  +  4x  +  y-3  =  0. 

3.  (0,  0),  2x2-22/2-2x  +  22/-l  =  0. 

4.  (4,  -2),  5^/2  +  \%y  4-4x4.  5  =  0. 

Find  the  poles  of  each  of  the  following  polars  with  Vespect  to  the  given  conic : 

5.  2x-y  =  0,  x2  +  8xy- 22/2 -12x4- 62/ -9  =  0. 

6.  x-32/4-2  =  0,  x2-f  1/2  _  2x4-42/ =  0. 

7.  x-f  2  2/-13  =  0,  3x2  +  82/2 -26x- 762/ -I- 231  =  0. 

8.  3x-22/-9  =  0,  3x2-4j/24-6x-242/-45  =  0. 


PEOBLEMS  263 

Find  the  equations  of  the  tangents  from  each  of  the  following  points  to  the 
given  conic : 

9.  (2,  3),  4x2  -  5X2/ +  2 2/2  +  3a;  -  22/ =  0. 

10.  (0,  1),  3x2  -  42/2  +  12x  =  0. 

11.  (1,  -2),  2x2 -22/2- 6x- 62/ -1  =  0. 

12.  (2,  4),  x2  +  2/2  -  6x  -  2 2/  +  5  =  0. 

13.  (2,  0),  5 2/2  +  4x  -  2  2/ -  8  =  0. 

14.  (-1,  -1),  3x2 +  8  2/2- 8X-122/  + 4  =  0. 

15.  Prove  that  the  polar  of  a  given  point  with  respect  to  any  one  of  the 
circles  x2  +  2/^  —  2  fcx  +  c2  =  0,  when  k  is  variable,  always  passes  through  a  fixed 
point  whatever  the  value  of  k. 

16.  T  is  the  pole  of  a  chord  PQ  of  the  parabola  y^  =  4px.  Prove  that  the 
perpendiculars  from  P,  T,  and  Q  upon  any  tangent  to  the  parabola  are  in 
geometric  progression. 

17.  If  P  is  any  point,  LM  its  polar  with  respect  to  any  central  conic,  C  the 
center  of  the  conic,  R  the  point  in  which  the  perpendicular  from  C  to  LM  meets 
LM,  and  S  the  point  in  which  the  perpendicular  from  P  to  LM  meets  the  axis 
of  the  conic,  prove  CR  ■  PS  =  b'^. 

18.  Prove  that  the  perpendicular  from  any  point  (xi,  2/1)  to  its  polar  with 
respect  to  any  central  conic  intersects  the  axis  of  the  conic  at  a  distance  e2xi 
from  the  center  of  the  conic. 

19.  Prove  that  if  in  any  conic  the  pole  of  the  normal  at  P  lies  on  the  normal 
at  Q,  then  the  pole  of  the  normal  at  Q  lies  on  the  normal  at  P. 

20.  If  Pi  and  P2  are  any  two  points,  and  C  the  center  of  a  conic,  show  that 
the  perpendiculars  from  Pi  and  C  to  the  polar  of  P2  are  to  each  other  as  the 
perpendiculars  from  P2  and  C  to  the  polar  of  Pi. 

21.  If  mi  is  the  slope  of  the  polar  of  a  point  Pi  with  respect  to  the  ellipse 

x2      w2 

1 =  1,  and  mg  is  the  slope  of  the  line  joining  Pi  to  the  center,  show  that 

o*      f>^       yi 

mirrii  = Find  the  similar  relation  for  the  hyperbola. 

a2 

22.  Prove  that  the  portion  of  the  axis  included  between  the  polars  of  two 
points  with  respect  to  a  parabola  equals  the  projection  on  the  axis  of  the  line 
joining  the  points. 

23.  Show  that  for  any  conic  section  the  polar  of  the  focus  is  the  directrix. 

24.  Where  is  the  polar  of  the  center  of  an  ellipse  or  hyperbola  with  respect 
to  that  curve  ? 

x2      ifl 

25.  In  the  ellipse 1-  —  =  1  find  the  equations  of  two  conjugate  diameters, 

o2      62 

one  of  which  bisects  the  chord  deteimined  by  the  upper  end  of  the  minor  axis 
and  the  right-hand  focus. 


264  TANGENT,  POLAR,  AND  DIAMETER 

26.  If  Pi  and  P2  are  the  extremities  of  any  two  conjugate  diameters  of  the 

ellipse 1-  ~  =  1,  prove  that  the  sum  of  the  squares  of  the  perpendiculars  drawn 

from  Pi  and  P2  to  the  major  axis  of  the  ellipse  is  equal  to  6^. 

27.  Show  that  there  can  be  only  one  pair  of  equal  conjugate  diameters  of  the 

ellipse h  ^  =  1,  namely  y  —  -x^y  — x. 

a^      6'^  a  a 

28.  Show  that  the  equation  of  any  ellipse  referred  to  its  equal  conjugate 

diameters  as  axes  is  x^  +  y^  = 

2 

29.  In  any  ellipse  show  that  the  diameters  parallel  to  the  lines  joining  the 
extremities  of  the  axes  are  conjugate. 

30.  One  diameter  of  the  ellipse \-  —  =  1  passes  through  the  upper  end  of 

the  right-hand  latus  rectum.    What  is  the  slope  of  the  conjugate  diameter  ? 

31.  What  must  be  the  relation  between  the  semiaxes  a  and  b  of  an  ellipse 
when  the  diameters  passing  through  the  upper  extremities  of  the  left-hand  latus 
rectum  and  the  right-hand  latus  rectum  are  conjugate  ? 

32.  Show  that  the  polar  of  any  point  on  a  diameter  of  a  central  conic  is 
parallel  to  the  conjugate  diameter. 

33.  Show  that  if  an  ellipse  and  an  hyperbola  have  the  same  axes  in  magni- 
tude and  position,  then  the  asymptotes  of  the  hyperbola  coincide  with  the  equal 
conjugate  diameters  of  the  ellipse. 

34.  Prove  that  tangents  at  the  ends  of  conjugate  diameters  of  an  hyperbola 
intersect  on  the  asymptotes. 

35.  Prove  that  the  straight  line  joining  the  ends  of  a  pair  of  conjugate  diam- 
eters of  an  hyperbola  is  parallel  to  one  asymptote  and  bisected  by  the  other. 

36.  If  an  hyperbola  has  a  pair  of  equal  conjugate  diameters,  prove  that  it  is 
an  equilateral  hyperbola. 

37.  Show  that  in  an  equilateral  hyperbola  conjugate  diameters  are  equally 
inclined  to  the  asymptotes. 

38.  Show  that  in  an  equilateral  hyperbola  all  diameters  at  right  angles  to 
each  other  are  equal. 

39.  Show  that  every  diameter  of  an  equilateral  hyperbola  is  equal  to  its 
conjugate. 

40.  Prove  that  the  tangents  at  the  ends  of  any  chord  of  a  conic  intersect  on 
the  diameter  which  bisects  the  chord. 

41.  The  chords  which  join  the  ends  of  any  diameter  to  any  point  of  the 
curve  are  called  supplemental  chords.  Prove  that  two  diameters  which  are 
parallel  to  any  pair  of  supplemental  chords  are  conjugate. 

42.  If  the  tangent  at  the  vertex  A  of  an  ellipse  cuts  any  two  conjugate 
diameters  produced  in  T  and  t,  show  that  AT ■  At  =  —  tfi. 


PROBLEMS  .  265 

43.  Show  that  if  any  tangent  meets  any  two  conjugate  diameters,  the  prod- 
uct of  its  segments  is  equal  to  the  square  of  the  lialf  of  tlie  parallel  diameter. 

44.  If  from  the  focus  of  an  ellipse  a  pei-pendicular  is  drawn  to  a  diameter, 
show  that  it  will  meet  the  conjugate  diameter  on  the  corresponding  directrix. 

45.  The  tangent  at  any  point  Pi  of  an  ellipse  cuts  the  equal  conjugate 
diameters  in  T  and  T^.  Show  that  the  triangles  TCP\  and  TiCPi  are  in  the 
ratio  CT'^  :  CTi\ 

46.  Show  that  the  product  of  the  focal  distances  of  any  point  of  a  central 
conic  is  equal  to  the  square  of  half  the  corresponding  conjugate  diameter. 

47.  Find  where  the  tangents  from  the  foot  of  the  directrix  will  meet  the 
hyperbola,  and  what  angles  they  will  make  with  the  transverse  axis. 

48.  Show  that  the  perpendicular  from  the  focus  upon  a  polar  with  respect  to 
an  ellipse  or  an  hyperbola  meets  the  line  drawn  from  the  center  to  the  pole  on 
the  corresponding  directrix. 


CHAPTER  XIII 
ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

148.  Definition.  Any  function  of  x  which  is  not  algebraic  is 
called  transcendental.  The  elementary  transcendental  functions  are 
the  trigonometric,  the  inverse  trigonometric,  the  exponential,  and 
the  logarithmic  functions,  the  definitions  and  the  simplest  properties 
of  which  are  supposed  to  be  known  to  the  student.  In  this  chapter 
we  shall  discuss  the  graphs  and  the  derivatives  of  these  functions. 

149.  Graphs  of  trigonometric  functions. 

Ex.  \.   y  =  sinx.  »    ■ 

The  values  of  y  are  found  from  a  table  of  trigonometric  functions.  In  plot- 
ting it  is  desirable  to  express  x  in  circular  measure  ;  e.g.  for  the  angle  180°  we 
lay  off  X  =  TT  =  3.1416.    When  x  is  a  multiple  of  tt,  y  =  0;  when  x  is  an  odd 

TT 

multiple  of  — ,  y  =  ±  1 ;  for  other  values  of  x,  y  is  numerically  less  than  1.    The 

graph  consists  of  an  indefinite  number  of  congruent  arches  alternately  above 
and  below  the  axis  of  x,  the  width  of  each  arch  being  tt  and  the  height  being  1 
(fig.  147). 

Y 


Fig.  147 


Tlie  curve  y  =  sin  x  may  be  con.structed  without  the  use  of  tables  by  a  method 
illustrated  in  fig.  148. 

Let  Pi  be  any  point  on  the  circumference  of  a  circle  of  radius  1  with  its 
center  at  C,  and  let  .40  be  a  diameter  of  the  circle  extended  indefinitely.  With 
a  pair  of  dividers  lay  off  on  A  0  produced  a  distance  OiVj  equal  to  the  arc  0P\. 
This  may  be  done  by  considering  the  arc  OPi  as  composed  of  a  number  of  straight 
lines  each  of  which  differs  unappreciably  from  its  arc.    From  ^i  draw  a  line 

266 


TRIGONOMETRIC  FUNCTIONS 


267 


perpendicular  to  A  0,  and  from  Pi  draw  a  line  parallel  to  A  0.  Let  these  lines 
intersect  in  Qi.  Then  iViQi  =  JfiPi  =  CPi  sin  OCPi.  But  CPi  =  1,  and  the  cir- 
cular measure  of  OCPi  is  OPi=  ONi.    If,  then,  we  take  ONi  =  x,  NiQi=  y. 


Fig.  148 

Qi  is  a  point  of  the  curve  y  =  sin  x.  By  varying  the  position  of  the  point  Pi  we 
may  construct  as  many  points  of  the  curve  as  we  wish.  The  figure  shows  the 
construction  of  another  point  Q2. 


Ex.  2.   y  =  a  sin  bx. 

TT  TT 

When  X  is  a  multiple  oi  ~,  y  =  0;  when  x  is  an  odd  multiple  of  —  ?  2/  =  ±  o ; 
b  26 

for  all  other  values  of  x,  y  is  numerically  less  than  a.    The  curve  is  similar  in 

its  general  shape  to  that  of  Ex.  1,  but  the  width  of  each  arch  is  now  - ,  and  its 
height  is  a.    Fig.  149  shows  the  curve  when  a  =  3  and  6  =  2. 


Fig.  149 


Ex.  3.    y  =  a  sin  {6x  +  c). 
c 


Place  X  = 


+  x',  y  =  y'. 


The  equation  then  becomes  y'  =  a  sin  6x'. 

The  graph  is  consequently  the  same  as  in  Ex.  2,  the  effect  of  the  term  +  c 
being  merely  to  shift  the  origin. 


268    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

Ex.  4.    y  =  a  cos  hx. 

This  may  be  written  y  =  asmlbx  -\ — |, 

which  is  a  cur\'e  of  Ex.  3.    Hence  the  graph  of  the  cosine  function  differs  from 
that  of  the  sine  function  only  in  its  position. 

Ex.  5.    y  =  sin  X  +  ^  sin  2  x. 

The  graph  is  found  by  adding  the  ordinates  of  the  two  curves  y  =  sin  x  and 
y  =  ^  sin  2  X,  as  shown  in  fig.  150. 


,-7/=^  sin  2x 


^      -^y^sin  x-t-jsin  231 
-s  y=sin  X 


Fig.  150 


Ex.  6.    y  =  sin 


y  =  0  when  -  =  Attt,  i.e.  when  x  =  -,  where  k  is  any  integer.    Hence  the 
X  k  " 

graph  crosses  the  axis  of  x  at  the  points  1,  ^,  ^,  \,  \,  etc.    Between  any  con- 
secutive two  of  these  points  y  varies  continuously  from  0  to  ±  1  and  back  to 


Fig.  151 


zero.    It  follows  that  as  x  approaches  0,  the  corresponding  point  on  the  graph 
oscillates  an  infinite  number  of  times  back  and  forth  between  the  straight  lines 


INVERSE  TRIGONOMETRIC  FUNCTIONS 


269 


y  =  ±1.    It  is  therefore  physically  impossible  to  construct  the  graph  in  the 

neighborhood  of  the  origin.    This  is  shown  in  fig.  151  by  the  break  in  the  cur%e. 

It  should  be  borne  in  mind,  however,  that  the  value  of  y  can  be  calculated 

for  any  value  of  x  no  matter  how  small.    E.g.  let  x  =  — - ;  then  -  = =  10  x 

125  X         12 

+  —  ir,  and  y  =  sin  —  =  sin  75°  =  .9659. 
12    '  12 

The  value  of  2/  is  not  defined  for  x  =  0,  and  the  function  is  discontinuous 

at  that  point. 


Ex.  7.    2/  =  tanx. 

When  X  is  a  multiple  of  tt,  y  =  0;  when  x  is  an  odd  multiple  of  - ,  y  is 

infinite,  in  the  sense  of  §§  11  and  68.    The  curve  has  therefore  an  iinlimited 

number  of  asymptotes  perpendicular  to  OX,  namely  x  =  ±  — »  x  =  ±  — »  •  •  • , 

which  divide  the  plane  into  an  infinite  number  of  sections,  in  each  of  which 
is  a  distinct  branch  of  the  curve,  as  shown  in  fig.  152. 


Fig.  162 


150.  Graphs  of  inverse  trigonometric  functions.  The  graphs  of 
the  inverse  trigonometric  functions  are  evidently  the  same  as  those 
of  the  direct  functions,  but  differently  placed  with  reference  to  the 
coordinate  axes.  It  is  to  be  noticed  particularly  that  to  any  value 
of  X  corresponds  an  infinite  number  of  values  of  y. 


'270    ELEMEIs^TARY  TliANSCEXDENTAL  FUoS'CTIONS 


Ex.  1.    2/  =  sin-la;. 

From  this  x  =  sin  y,  and  we  may  plot  the  graph 
by  assuming  values  of  y  and  computing  those  of  x 
(fig.  153). 


Fig.  1o3 


151.  Limits  of 


Fig.  154 

Ex.  2.    y  =  tan-ix. 

Then  x  =  tan  j/,  and  the  graph  is  as  in  fig.  154. 
These  curves  show  clearly  that  to  any  value  of  x 
corresponds  an  infinite  number  of  values  of  y. 


sin^ 


and 


1  —  cos  A 


In  order  to  apply  the 


h 


h 


methods  of  the  differential  calculus  to  the  trigonometric  functions,  it 

is  necessary  to  know  the  limits  approached  by  — : —  and 
as  h  approaches  zero  as  a  limit,  it 
being  assumed  that  h  is  expressed 
in  circular  measure. 

1.  Let  AOB  (fig.  155)  be  the 
angle  h,  r  the  radius  of  the  arc  AB 
described  from  0  as  a  center,  a  the  ^ ' 
length  of  AB,p  the  length  of  the  per- 
pendicular BC  from  B  to  OA,  and 
t  the  length  of  the  tangent  drawn 
from  B  to  meet  OA  produced  in  D.  Fig.  155 


^^ 

p 

V' 

^^^^^1^^ 

c 

^1  V 

-^ 

1  / 

^^^ 

1  / 

■v^ 

/  / 

■^^ 

/  / 

'^^v 

// 

^v 

,. 

/ 

^^. 

/ 

B' 


CERTAIN  LIMITS  271 

Eevolve  the  figure  on  OA  as  an  axis  until  B  takes  the  position 
B'.    Then  BCB'  =2p,  BAB'  =  2  a,  B'D  =  BD.    Evidently 

BD  +  DB'  >  BAB'  >  BCB', 
whence  t>  a>  p. 

Dividing  through  by  r,  we  have 

tap 
T       r       r 
that  is,  tan  h>  h>  sin  h. 

Dividing  by  sin  h,  we  have 


cos  h      sin  h 
or,  by  inverting,  cos  h  <  — - —  <  1, 

Now  as  h  approaches  zero,  cos  h  approaches  1.    Hence  > 

which  lies  between  cos  h  and  1,  must  also  approach  1 ;  that  is, 

-r .      sin  h      . 
Lim  ——  =  1. 

1 cos  Jb 

2.  To  find  the  limit  of  — — - — -  »  as  h  approaches  0,  we  proceed 

as  follows : 

,             ;        2sin^J      sin'^^       ,  /sin^^ 
1  —  cos  h  ^ 2  ^ 2  ^  h    2 

h  h  h         ^\    Ik 

2  \   2 

.    h 
sm- 

Now  as  h  approaches  zero  as  a  limit,  — -^  approaches  1,  as 
just  shown,  and  therefore  -   ■         j  approaches  zero,  by  2,  §  94. 

Therefore  Lim =  0. 

A±0  h 


272    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

152.  Differentiation  of  trigonometric  functions.  The  formulas 
for  the  differentiation  of  trigonometric  functions  are  as  follows, 
where  u  represents  any  function  of  x  which  can  be  differentiated : 

d     .  du 

—-  sm u  =  cos u—-}  .  (1) 

dx  dx 

d  .       du  ,-, 

—  cos  u=  —  sm  u-—>  (2) 
dx                            dx 

d     .  2       <^^  /ov 

—  tan -i^  =  sec  M  — -  >  (3) 
dx                         dx 

d       .  ^    du  ..^ 

-—  COtW  =  —  CSC  M-—>  (4) 

dx  dx 

d  du 

—  sec  u  =  sec  u  tan  u-—>  (5) 
dx                                 dx 

d  .     du  _^ 

-— cscw  =— cscw  cotM  — -•  (6) 

dx  dx 

1.  By  ( 0>  S  96,     --  sm  %  =  -—  sm  w  •  --  • 
dx  du  dx 

To  find  — -  sin  u,  we  place  y  =  sin  u. 
du 

Then  if  u  receives  an  increment  Aw,  y  receives  an  increment  A?/, 

where  /         *    \         » 

.  .    ,         *    s        •  o        /         Aw\    .    Am 

Ay  =  sm  {u  +  Azt)  —  sm  u^  1  cos  i  u  +  -—  I  sm  —  > 

the  last  reduction  being  made  by  the  trigonometric  formula 

.    ,       „        a  -\-h    .    a  —  b 
sm  a  —  sm  o  =  2  cos  — - —  sm  — - — 
2  2 

Then  we  have  *  a 

.    Au  .    Aw 

V  sm  — —  ,  sm  -— 

Ay      ^^        /        Au\         2  /     ,  Au\         2 

— ^  =  2  cos  I  u  +  — —    =  cos   u  +  — — 

Au  \2/Aw  V2/Aw 

2 

Let  Au  approach  zero.    By  2,  §  94, 

.    Aw 
sm— - 

Lim  —  =  Lim  cos  (  w  +  — -  )  Lim  — r ' 

Aw  \         2  /  Aw 


DIFFERENTIATION     .  273 


But  Lini  — ^  =  -^  ,  Lim  cos  (  m  +  — -  )  =  cos  u,  and  Lim 
Alt      du  \  2  / 


=  1(§151).  2 

Hence  -—  sin  w  =  cos  u 

du 

d    .  du 

and  —- sm  u  =  cos  u  - — 

dx  dx 

2.  To  find  —  cos  u,  we  write 
dx 

■    i-^ 
cos  w  =  sin  I  —  —  t* 


Then  -—  cos  w  =  -7-  sin  \—  —  ^t 

dx  dx       \l 

/tt         \  d  (it 


=°°%2-'7rf:;i2-'''  <''y<'» 


"K        \du 

du 
=  —  sm  w  ^-  • 
dx 


3.  To  find  —  tan  u,  we  write 

sinw 

tan  u  = 

cos  w 


^,  d  ^  d   smu 

Then  -r  tan  -26  =  -^ 

aa;  dx  cos  -it 


cos  u  —  sin  u  —  sm  u  —-  cos  u 


ax                            ax 

(hvi^)  ^^^) 

cos^u 

\"j  \  />  0       / 

.  0  K  du 

(cos^u  +  siw'u)  — 

cos^u 

(by  (1)  and  (2)) 

0     du 
=  sec  w  -7-- 
dx 

274    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS     . 

4.  To  find  -T-  cot  u,  we  write 

ax 

cos  u 

cot  u  — 

sin  w 

^,  d      ^  d   cosu 

Then  --cotu  = : — 

ax  dx  sinw 

d  d    . 

sin  u  —  cos  u  —  cos  ti  —  sin  u 

= 3i^.^ (by  (5).  §  96) 

^-sin^^-cos^.^  (by  (1)  and  (2)) 

=  —  CSC  M  •  — -• 

dx 

5.  To  find  — -  sec  u,  we  write 

dx 

sec  u  = =  (cos  u)  \ 

cosu 

Then  —  sec  u=  —  (cos  ti)~  ^  -—  cos  u  (by  §  97) 

dx  dx 


_  sin  u  du 
cos^w  dx 

du 
=  sec  it  tan  u  — —  ■ 


(by  (2)) 


6.  To  find  —  CSC  u,  we  write 
dx 

CSC  w  =  -; =  (sm  u)    . 

suiti 

Then  —  esc  ?t  =  —  (sin  u)~^  — -  sin  u  (by  §  97) 

aa?  dx 

=  —  CSC  M  cot  w  —  •  (by  (1)) 

ax 

Ex.  1.    y  =  tan  2  x  —  tan^  x  =  tan  2  x  —  (tan  x)^. 

—  =  sec2  2  X      (2  x)  —  2  (tan  x)  —  tan  x 
dx  dx  dx 

=  2  sec2  2  X  —  2  tan  x  sec^  x. 


DIFFERENTIATION 


275 


Ex.  2.    A  particle  moves  in  a  straight  line  so  that 

s  =  A;  sin  bt, 
where  t  =  time,  s  =  space,  and  b  and  k  are  constants.    Then 

velocity  =  w  =  —  =  bk  cos  bt, 
dt 

acceleration  =  a  =  — '-  =  — b^k  sin  bt  =  —  6%, 

dt^  ' 

force  =  F=  ma  =  —  mb^s. 

Let  0  be  the  position  of  the  particle  when  ^  =  0,  and  let  OA  =  +  k  and 
OB  =  —  k.  Then  it  appears  from  the  formulas  for  s  and  v  that  the  particle 
oscillates  forward  and  backward  between  B  and  A.    It  describes  the  distance 

_  2  77" 

OA  in  the  time  — ,  and  moves  from  B  to  A  and  back  to  B  in  the  time 

26  -6 

The  formula  F  =  —  mb^s  shows 
that  the  particle  is  acted  on  by  a 
force  directed  toward  O  and  pro- 
portional to  the  distance  of  the 
particle  from  0. 

The  motion  of  the  particle  is 
called  simple  harmonic  motion. 

Ex.  3.  A  wall  is  to  be  braced 
by  means  of  a  beam  which  must 
pass  over  a  lower  wall  6  units 
high  and  standing  a  units  in  front 
of  the  first  wall.  Required  the 
shortest  beam  which  can  be  used. 

Let  AB  =  I  (fig.  156)  be  the  beam,  and  C  the  top  of  the  lower  wall. 
Draw  the  line  CD  parallel  to  OB  and  let  EEC  =  0.    Then 


that  is,  when 


1  =  BC  +  CA 

=  EC  esc  0  +  DC  sec  0 
=  bcsc0  +  a  sec  9 

—  =  —  6  CSC  ^  cot  ^  +  a  sec  tf  tan  0 
do 

_  asin^e  -  bco^^0 

Si\vfi  0  cos"^  6 

—  =  0,     when    a  sin^  0=b  cos^  5, 
de 

bi 
tan  0  =  —  • 


270    ELEMENTARY  TRAKSCENDEKTAL  FUNCTIONS 

When  e  has  a  smaller  value  than  this,  a  sin^d  <  h  cos^^,  and  when  d  has  a 

larger  value,  a  sin^tf  >  b  cos^tf.    Hence  Hs  a  minimum  when  tan  d  ■=  —   Then 

o* 

i  =  6  CSC  ^  +  a  sec  tf 


6  Vg^  +  6^      g  VaJ  + 


6*  g* 

=  (a^  +  b^)^. 

153.  Differentiation  of  inverse  trigonometric  functions.    The 

formulas  for  the  differentiation  of  the  inverse  trigonometric  func- 
tions are  as  follows : 

1.  —&\n~^u=      rr^  —  >  when  sin"'w  is  in  the  first  or  the 

VI  — w  fourth  quadrant; 

= —>  when  sin~'w  is  in  the  second  or 

V 1  —  w  ^j^  third  quadrant. 

2.  --T-  cos~*w  = —  >  when  cos"'w  is  in  the  first  or  the 

^  ~  ^  second  quadrant ; 

>  when  cos~'w  is  in  the  third  or  the 


fourth  quadrant. 


Vl  — tt'^  dx 

ax  1  +  u   ax 

.     d      ^  ,  1      du 

4  --  cot-^u  =  — -—  • 

ax  1  +  u   ax 

5.  — -sec"^^  =  - —  —  >  when  sec" 'w  is  in  the  first  or  the 

dx  .       uVu'  - 1  dx  ^j^.^^  quadrant ; 

—  >  when  sec~^ic  is  in  the  second 


wVit       IX         ^^  ^-^^  fourth  quadrant. 


6,  — csc~'w= y^^z^  — >  when  esc  ^w  is  in  the  first  or 

dx  uy/u'  - 1  dx         ^-^^  ^j^.^^^  quadrant ; 

— ,  when  csc"'?^  is  in  the  second  or 


y/u^  —  l  dx 


uy/u 


the  fourth  quadrant. 


DIFFERENTIATION  277 

The  proofs  of  these  formulas  are  as  follows : 

1.  If  y  =  sin"^w, 
then                                     sin  y  =  u. 

Hence  ^^^^'^~'J~'  (^"7  §  152) 

dy         1     du 
or  •        _£  = 

dx      cosy  dx 

But  cos  y  =  Vl  —  u^,  when  y  is  in  the  first  or  the  fourth  quad- 
rant, and  cosy  =— vl  — w^  when  y  is  in  the  second  or  the  third 
quadrant. 

2.  If  y  =  cos'^w, 
then                                     cos  y  =  u; 

dy      du 
whence  —siny-f-  = 


or 


lA/Jb  (a/Ju 

dy  _         1      du 

dx  sin  y  dx 


But  sin 3/  =  +  Vl—  u^  when  y  is  in  the  first  or  the  second 
quadrant,  and  siuy  =  — Vl  — ti"-^  when  y  is  in  the  third  or  the 
fourth  quadrant. 

3.  If  y  =  tsinr^u, 
then                                    tan  y  =  u. 

Hence 
whence 

4.  If 
then 


Hence 

whence 


2     dy 

du . 
dx 

dy  _ 

1 

du 

dx 

1+' 

u^  dx 

y  = 

■■  cot" 

'u. 

coty  = 

u. 

2     <^V 

du 

dx 

dy  _ 

1      ( 

dx 

1 

+  w'  ( 

du 


278    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 


5.  If 

y  =  sec"'tf, 

then 

sec  y  =  u. 

Hence 

dy      du 
secytany-f-  =  — , 
dx      dx 

whence 

dy              1 

du 
dx      sec  y  tan  y  dx 


But  sec  y  =  u  and  tan  y  =  +  ^u^  —  1  when  y  is  in  the  first  or 
the  third  quadrant,  and  tan  y  =  —  ^li^  —  1  when  y  is  in  the  second 
or  the  fourth  quadrant. 


b.  if 

y  =  csc"'w, 

CSC  y  =  u. 

Hence 

dy      du 

—  CSC  y  cot  y  -T-  =  -rr-y 

dx      dx 

whence 

dy                  1          du 

dx          CSC  y  cot  y  dx 

But  CSC  y  =  u  and  cot  y  =  +  v  ?t^  —  1  when  y  is  in  the  first  or 
the  third  quadrant,  and  cot  y  =  —  vw^  —  1  when  y  is  in  the  second 
or  the  fourth  quadrant. 


Ex.  1.    y  =  sin-i  Vl  —  x^,  where  y  is  an  acute  angle. 


^      Vl  -  (1  -  a;2)    <i^  Vl-x2 


This  may  also  be  clone  by  noticing  that  sin-^  v  1  —  x^  =  cos-^x. 

Ex.  2.    The  example  of  §  107  may  be  solved  by  drawing  a  straight  line  from 
S  to  0  (fig.  125),  denoting  the  angle  YOS  by  6  and  the  subtended  arc  by  s. 


Then  s  =  ad, 

rL.S  =  2tan-i  

OL  a 


and  6  =  1-  YLS  =  2  tan-i =  2  tan-i^ 


Hence 


and 


s  =  2  a  tan-'  — , 
a 

1 

ds      -           a 

V  =  —  =  2a 

dt.                   x'{ 

dxi        2  a'^c 
dt       a2  +  xl 

EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    279 


154.  The   exponential   and   the   logarithmic    functions.     The 


equation 


2/  =  a" 


defines  y  as  a  continuous  function  of  x,  called  the  exponential  func- 
tion, such  that  to  any  real  value  of  x  corresponds  one  and  only 
one  real  positive  value  of  y.  A  proof  of  this  statement  depends 
upon  higher  mathematics,  but  the  student  is  already  familiar  with 
the  methods  by  which  the  value  of  y  may  be  computed  for  simple 
values  of  x.    li  x  =  n,  an  integer,  y  is  determined  by  raising  a  to 

P 
the  nth.  power  by  multiplication.    If  a  is  a  positive  fraction  —  >  y  is 

the  g-th  root  of  the  ^th  power  of  a.  If  ic  is  a  positive  irrational  num- 
ber, the  approximate  value  of  y  may  be  obtained  by  expressing  x 
approximately  as  a  rational  number.    If  a?  =  0,  3/  =  a"  =  1.    Finally, 

if  x  =  —  m,  where  m  is  any  positive  number,  3/  =  a,""*  =  —  • 

Practically,  however,  the  value  of  a^  is  most  readily  obtained  by 
means  of  the  inverse  function,  the  logarithm  ;  for  if 


then 


a;  =  log„y. 


When  a  =  10,  tables  of  log- 
arithms are  readily  accessible. 
Suppose  a  is  not  10,  and  let  h 
be  such  a  number  that 


10*  =  a, 


I.e. 


and 


h  =  log,oa. 

Then  we  have 

y  =  a^={lQy  =  W. 

Hence         hx  =  log^^^y, 

^      log,„y      log,o2/ 


logio^ 


Fig.  167 


Ex.  1.    The  graph  oi  y  =  log(i.6)X  is  shown  in  fig.  157. 

It  is  to  be  noticed  that  the  curve  has  the  negative  portion  of  the  y  axis  for 
an  asymptote,  and  has  no  points  corresponding  to  negative  values  of  z. 


280    ELEMENTARY  TRAKSCENDEXTAL  FUXCTIONS 


Ex.  2.    The  graph  oi  y  =  (1.5)*  is  shown  in  fig.  158. 


Fig.  158 


155.  The  number  e.    In 

the  theory  and  the  use  of  the 
exponential  and  the  logarith- 
mic functions,  an  important 
part  is  played  by  a  certain 
irrational  number,  commonly 
denoted  by  the  letter  e.  This 
number  is  defined  by  an 
X  infinite  series,  thus : 


It  will  be  shown  in  the  second  volume  that  this  series  converges; 
i.e.  that  the  greater  the  number  of  terms  taken  the  more  nearly 
does  their  sum  approach  a  certain  number  as  a  limit.  Assuming 
this,  we  may  compute  e  to  seven  decimal  places  by  taking  the  first 
eleven  terms.    There  results 

e=  2.7182818.  ••. 


When  y  =  e^,x  is  called  the  natural  or  Napierian  logarithm  of  y. 
The  student  will  discover  as  he  proceeds  with  his  study  that  the 
use  of  Napierian  logarithms  in  theoretical  work  causes  simpler 
formulas  than  would  arise  with  the  use  of  the  common  logarithm. 
Hence,  in  theoretical  discussions,  the  expression  logic  usually  means 
the  Napierian  logarithm.  On  the  other  hand,  when  the  chief  inter- 
est is  in  calculation  of  numerical  values,  as  in  the  solution  of  tri- 
angles, logic  usually  means  logjoX  In  this  hook  we  shall  use  log x 
for  log^x. 

Tables  of  values  of  log^a:  and  (f  are  found  in  many  collections 
of  tables,  and  may  be  used  in  finding  the  graphs.  It  is  evident, 
however,  that  the  graphs  will  not  differ  in  general  shape  from  those 
in  Exs.  1  and  2  of  §  154. 

We  give  the  graphs  of  certain  other  functions  which  involve  e 
and  present  other  points  of  interest. 


GRAPHS 


281 


Ex.  1.    y  =  e-=^. 

The  curve  (fig.  159)  is  symmetrical  with  respect  to  OY  and  is  always  above 
OX.  When  x  =  0,y  =  1.  As  x  increases  numerically  y  decreases,  approaching 
zero.    Hence  OX  is  an  asymptote. 

Y 


Fig.  159 


Ex.  2.    y 


-(ea  +  e  »). 


This  is  the  curve  (fig.  160)  made  by  a  string  held  at  the  ends  and  allowed  to 
hang  freely.    It  is  called  the  catenary. 


Ex.  3.    y  =  e~°^sin6x. 

The  values  of  y  may  be  computed  by  multiplyi:ig  the  ordinates  of  the  curve 
y  _  e-ax  by  the  value  of  sin  bx  for  the  corresponding  abscissas.  Since  the  values 
of  sin6x  oscillate  between  ±  1,  the  value  of  e-'"sin6x  cannot  exceed  those  of 
e-°^.  Hence  the  graph  lies  in  the  portion  of  the  plane  between  the  curves 
y  =  e-«^  and  ?/  =  -  e-"^.    When  x  is  a  multiple  of  -  ,  y  is  zero.    The  graph 


282    ELEMENTAEY  TIIANSCEKDE:N^TAL  FUNCTIOI^S 

therefore  crosses  the  axis  of  x  an  infinite  number  of  times.    Fig.  161  shows  the 
graph  when  a  =  1,  6  =  2  7r. 


1 

\ 

1  \  ^\ 

/\^^ ~}^--<z~ 

0 

i       \.      y2 ::: 

-1 

/"" 

Fig.  161 

Ex.  4.    y  =  e^. 

When  X  approaches  zero,  being  positive,  y  increases  without  limit.    When  x 
approaches  zero,  being  negative,  y  approaches  zero;    e.g.  when  x=  yoV(J' 

y  =  el"*'",  and  when  «  =  —  y^Vo'  y  =  er'^'**^  ~  "wm'    ^^®  function  is  therefore 
discontinuous  for  x  =  0. 


3 

T 

; 

L 

\ 

0 

Fig.  162 


The  line  y  =  1  is  an  asymptote  (fig.  162),  for  as  x  increases  without  limit, 
being  positive  or  negative,  -  approaches  0  and  y  approaches  1. 


CERTAIN  LIMITS 


283 


10 


Ex.  5.    y  = 

L 

1  +  e^ 
As  X  approaches  zero  positively,  y  approaches  zero.    As  x  approaches  zero 
negatively,  y  approaches  10.    As  z  increases  indefinitely,  y  approaches  6. 
The  curve  (fig,  163)  is  discontinuous  when  x  =  0. 


10 


Fig.  163 


156.  Limits  of  (l  +  hf  and 


e^'-l 


In  obtaining  the  formulas 


for  the  differentiation  of  the  exponential  and  the  logarithmic  func- 
tions it  is  necessary  to  know  certain  limits,  the  rigorous  derivation 
of  which  requires  methods  which  are  too  advanced  for  this  book. 
We  must  content  ourselves,  therefore,  with  indicating  somewhat 
roughly  the  general  nature  of  the  proof. 

1.  We  require  the  limit  of  (1  +  A)*  as  h  approaches  zero.    We 
1. 
begin  by  expanding  (1+  ^)*  by  the  binomial  theorem  and  making 

certain  simple  transformations ;  thus : 


a+.)i=i+i..+i6: 


h'  + 


¥ 


A«  + 


_      1     (1-;^)     (i-;^)(i-2A) 


■-^^i-*-|^+ii  + 


-hB, 


284    ELEMENTAEY  TRANSCENDENTAL  FUNCTIONS 

where  7^  represents  the  sum  of  all  terms  involving  h,  h^,  h^,  etc. 
Now  it  may  be  shown  by  advanced  methods  that  as  h  approaches 
zero  R  also  approaches  zero,  and  at  the  same  time 

i  +  T  +  g  +  @  +  -- 

approaches  e.    Hence  i 

Lim(l+A)A  =  e. 

A=  0 

e*— 1 
2.  We  require  the  limit  of  — - —  as  h  approaches  zero. 

h 

Let  us  place  e*  —  1  =  k, 

where  evidently  k  is  a  number  approaching  zero  as  h  approaches 

zero.    Then 

e*  =  1  +  ^,     whence     A  =  log  (1  +  k). 

Then  we  have 

g"-l_         k         _  1  _  1 

h     -log(l4-^^)-Log(i+Z:)"log(l  +  >t)^' 

K 

1 

Now  as  h  approaches  zero  k  approaches  0,  and  (1  +  ^)*  approaches 

e  by  the  previous  proof.    Hence  log(l  +  ky  approaches  log  e,  which 

is  1.    Therefore  e*— 1 

Lim  — - —  =  1. 

»=o      h 

157.  Differentiation  of  exponential  and  logarithmic  functions. 
The  formulas  for  the  differentiation  of  the  exponential  and  logarith- 
mic functions  are  as  follows,  where,  as  usual,  it  represents  any  func- 
tion which  can  be  differentiated  with  respect  to  x,  log  means  the 
Napierian  logarithm,  and  a  is  any  constant: 

dx  dx 

d  1  I  du 

-7-  log  W  = ;-  •  (2) 

dx  u  dx 

—  a-^a-loga^^.  (3) 

dx  ""     dx  ^  ' 

d  ,  _  log„g  du  . 

dx  u     dx 


DIFFERENTIATION  285 

dx  du       dx 

To  find  —  e",  place  y  =  e".    Then  if  w  receives  an  increment  Aw, 
du 

y  receives  an  increment  Ay,  where 

Ay  =  e"  + ''"  —  e"  =  e"  (e^"  —  1). 


Then 

Aw      ""      Aw 

Now  let  Aw 

approach  zero. 

By  §  94, 

Lim  — ^  =  6"  Lim  — -^ 

Aw                     Aw 

But 

-..Ay      ^y       c?    „ 
Lim  -^  =  -/■  =  —  e", 
Lu      du      du 

and 

Lim  — - —  =  1. 

Aw 

Therefore 

—  e"  =  e", 
ait 

and 

d    „       ^du 

2.  If 

y  =  logw, 

then 

e>'  =  u. 

Hence 

ydy  _du 
dx      dx 

whence 

dy      1  du  _ldu 
dx      e"  dx      u  dx 

by  2,  §  156 


by(l) 


3.  Let  y=a". 

Then  it  is  always  possible  to  find  a  quantity  b  such  that 
a  =  e\ 
whence  ^  =  log  a. 


286    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 


Then 

y  =  {e")"  =  e"\ 

and 

dx          dx 

dx 

=  (log«)».|. 

4  If 

y  =  log^-?^, 

then 

a«  =  u, 

and 

„  T         dy      du . 
a^  log  a  -^  =  — -  > 
dx      dx 

whence 

dy         1       1    du 
dx      log  a    u    dx 

But  if 

log  a  =  h, 

a  =  e^, 

1 

whence 

(^  =  e, 

and  therefore 

\  =  log.*. 

Hence  ■  ^^\og^dv. 

dx         u     dx 


Ex.1,    y  =  log(x2_4x  + 6). 

iZj/  2x  —  4 


by(i) 


by  (3) 


dx     x2  _  4  x  +  6 
Ex.  2.    y  =  e-=^.  _ 

:  ■  da; ■ 

Ex.  3.    2/  =  e-'^cosftx. 

dy  .d,,  d,,,  ,,  ., 

-^  =  cosox  —  (e-'«^)  +  e-"^ —  (cosox)  =  —  ae-°^cos6x  —  oe-»*sinc>x. 
dx  dx  dz    , 


EXPONENTIAL  FUNCTION  287 

158.  An  important  property  of  the  exponential  functions  is 
expressed  in  the  following  theorem :  If  the  rate  of  change  of  a 
function  is  proportional  to  the  value  of  the  function,  the  function 
is  an  exponential  function. 

Let  -^  =  ay. 

Then  l^  =  a. 

y  dx 

Hence  log  y  =  ax  +  c^, 

or  y  =  g^^^+'^i  =  e'^'e'^  =  ce'^. 

Ex.  Let  p  be  the  atmospheric  pressure  at  the  distance  h  above  the  surface 
of  the  earth  and  p  the  density  of  the  air.  We  will  assume  that  the  density  is 
proportional  to  the  pressure.  Then  if  po  and  po  are  the  density  and  the  pressure 
respectively  at  the  surface  of  the  earth, 

Po      Po' 

whence  p  =  —  •  p. 

'^      Po 

Let  now  the  height  h  be  increased  by  a  distance  Ah.  The  pressure  will 
be  increased  by  an  amount  Ap,  where  —  Ap  is  equal  to  the  weight  of  a  column 
of  air  standing  on  a  base  of  unit  area  and  having  a  height  Ah.  If  p  is  the  density 
at  the  height  h  and  p  —  Ap  the  density  at  the  height  h  +  Ah,  it  is  evident  that 
the  weight  of  this  column  of  air  lies  between  (p  —  Ap)  Ah  and  pAh ;  that  is, 

(p  —  Ap)  Ah<  —  Ap  <  pAh, 

Ap 
whence  p  —  Ap  < r  <  P- 

Ah 


Taking  the  limit,  we  have 

dp      ...    .^Ap  PO 

■— z=Limit— =- p  =  - ^p. 
dh  Ah  Po 

-Bih  ...      . 

Therefore  p  =  ce  po  .  .•..•■... 

_?o  A                                                                            ..... 
•Since  when  it  =  0,  e  i'o    =1  and  p  =  po,  it  follows  that  c  =  p^.'    "' • 

Hence  p  =  poe   I'o  , 

,       Po,     Po 
/.  =  -log^. 


288    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

159.  Sometimes  the  work  of  differentiating  a  function  is  sim- 
plified by  first  taking  the  logarithm  of  the  function  and  then 
applying  the  formulas  of  this  article. 


Ex.  1.    Let 


=  Vr 


X2 


+  X2 


- 


Hence 


and 


+  X''' 
=  ^l0g(l-x2)-^.l0g{l+«2). 

1  dy  _  X  X 

y  dx~      I  -  x^      1  +  X* 
-2x 
~  (1  -  x-2)  (1  +  x2) 
dy  _  —2xy 

dx~  (1-  cc2)  (1  +  a;2) 


-2x  l-x2 


x'^)  \1 


(1  -  x2) (1  +  x-i)  \l  +  x3 
-2x 


(l  +  x2)V'l-x< 

This  method  is  especially  useful  for  functions  of  the  form  u", 
where  u  and  v  are  both  functions  of  x.  Such  functions  occur 
rarely  in  practice,  and  cannot  be  differentiated  by  any  of  the 
formulas  so  far  given.  By  taking  the  logarithm  of  the  function, 
however,  a  form  is  obtained  which  may  be  differentiated. 

Ex.  2.    Let  y  =  x«'°^. 

Then  log  y  =  log  (x""  ^) 

=  sin  X  •  log  X. 

Therefore  — -  =  (sin  x)  -  +  cos  x  ■  log  z, 

y  dx  X 

dy 
and  -^  =  x"° ^-  J  •  sin  X  +  x*'°^ cos  x  •  log x. 

dx 

160.  Hyperbolic  functions.  Certain  combinations  of  exponential 
functions  are  called  hyperbolic  functions.  In  their  names  and 
properties  they  are  analogous  to  the  trigonometric  functions,  but 
the  reason  for  this  cannot  be  shown  at  present.  The  fundamen- 
tal hyperbolic  functions  are  the  hyperbolic  sine  (sinh),  the  hyper- 
bolic cosine  (cosh),  the  hyperbolic   tangent  (tanh),  the   hyperbolic 


HYPERBOLIC  FUNCTIONS 


289 


cotangent  (coth),  the  hyperbolic  secant  (sech),  and  the  hyperbolic 
cosecant  (cosech),  defined  by  the  equations 


sinha;  = 
cosh  X  = 
tanh  X  = 
coth  a;  = 
sech  X  = 
cosech  X  = 


e^—e-^ 

2      ' 

e^+e-^ 

2      ' 

sinha; 

e^—e 

X 

cosh  a: 

e^-f-e" 

X 

cosh  X 

e^^+e- 

-X 

sinh  X 

e* —  e~ 

X 

1 

2 

cosh  a? 

e^'+e- 

X 

1 

2 

sinh  a?      e^  —  e" 


Fig.  164 


Fio.  105 


The  graph  of  sinh  x  is  given  in  fig.  164,  that  of  cosh  x  in 
fig.  165,  and  that  of  tanh  x  in  fig.  166. 


290    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

Relations  between  hyperbolic  functions  may  be  derived  by 
expressing  each  in  terms  of  the  exponential  functions.  The 
student  may  in  this  way   prove  the  following  relations: 

cosh'^a;  —  sinh^a;  =  1, 

tanh^ic  +  sech^a?  =  1, 

coth'*  X  —  cosech^  x  =  \, 

sinh  (a?  ±  y)  =  sinha?  coshy  ±  cosh  a;  sinhy, 

cosh  {x  ±  y)  =  cosh  X  cosh  y  ±  sinh  x  sinh  y, 

^     1  /     ,     X       tanhaj±tanhy 
tanh(a;±y)—  ^ 


Fig,  166 

The  derivatives  of  the  hyperbolic  functions  are  readily  obtained 

by  differentiating  the  equations  which  define  them.    We  have  in 

this  way :  ,  , 

•'  a     .  ,  ,      an 

-T—  smh  u  =  cosh  u-—> 
ax  ax 

d       ,  .  ,      du 

-r-  cosh  u  =  smh  u  — - , 
dx  dx 

d  ^     ,  1 2    du 

-—  tanh  u  =  sech  w  -— , 
dx  dx 

d      ^,  ,  ^    du 

-r- coth u=—  cosech  u—-, 
dx  ax 

-T-  sech  u  =—  sech  u  tanh  u  — - , 
dx  dx 

——  cosech  u  =  —  cosech  u  coth  u  — -• 
dx  dx 


INVERSE  HYPERBOLIC  FUNCTIONS  291 

161.  Inverse  hyperbolic  functions.    If 

X  =  sinli  y, 
then  '  y  =  smh"^a;, 

called  the  inverse  hyperbolic  sine  of  x. 

This  function  may  be  expressed  as  a  logarithm  as  follows : 

We  have                   y  =  sinh~  ^  x, 
and  X  =  sinh  y  =  — 

Placing  e~^=  —  and  clearing  of  fractions,  we  have 
e^''—2xe''=l. 

Treating  this  as  a  quadratic  equation  in  «",  we  have 

e^=x±Va^+l; 

but  since  we  know  that  for  any  real  value  of  y,  e"  is  positive,  we 
discard  the  minus  sign  before  the  radical  and  have 


y  =  sinh"'  ic  =  log  {x  +  vV+T). 
In  the  same  manner,  the  student  may  prove  the  following : 


cosh~'a;  =  log  {x  ±Vic^—  1) 
=  ±log 

tanh~'a;  =  |  log 


=  ±  log  (x  +  ^y?  —  1)' 


coth"'a;=  |  log 


\—x 

x  +  1 
x  —  1 


,      l±Vl-a;*-^      _^,      l  +  Vl-ar' 
sech"'  a?  =  log =  ±  log 

^  X  X 

cosech"^'  ic  =  log • 


292    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 

The    derivative    of   the    inverse    hyperbolic    functions    can  be 

obtained  by  differentiating  the  expressions  just  obtained,  or  by 

proceeding   in    the   same  manner  as  in   §  153.     In  either  way 

we  find:  ,  .         , 

a     .  ,    ,  1       du 

—  sinh    u  =■—= — J 

dx  V  M^  -\-l  dx 

d       ,  _,  ,         1       du 

—  cosh  '■u  =  ±—:  —-, 

dx  ^u'-l  dx 

d  ^     ,  _,  1      du 

-— tanh  ^u  =  - r-— , 

dx  1  —  u  dx 

d      ^.  _,  1      du 

--coth    u=- --—, 

dx  1  —  udx 

d       ,    ,  1         du 

—  sech  ^u  =  zf  —  — , 

dx  u\l—  u^  dx 

d           ,  _,                    1         du 
—  cosech    u  = 

dx  uV  1+  u^  dx 

Ex.    Consider  the  motion  of  a  particle  of  unit  mass  falling  from  rest,  and 

impeded  by  a  force  proportional  to  the  square  of  its  velocity.    The  total  force 

acting  on  the  particle  is  then  g  —  kv'^,  where  g  is  the  acceleration  due  to  gravity, 

and  A;  is  a  constant.    Hence 

—  =  g-kv^; 
at 

1       dv 

whence  =  1, 

g  -  kv^  dt 

1        1         dv      , 

or  =  1. 

ff    ^      k  .  dt 

9 

To  bring  this  under  one  of  the  known  formulas  of  differentiation  we  will 
place 


whence 


dv  _     jg  du 


We  have,  therefore,  — - — =  1 ; 


whence  — — =tanh-i  u  z=t  +  c, 


Vkg  i-u^  dt 


1 


{k 
tanh-i-%/-i'  =  <  +  f. 


TRANSCENDENTAL  EQUATIONS 


293 


But  since  the  body  falls  from  rest,  when  t  =  0,  v  =  0;  therefore  c  =  0. 
The  equation  may  be  written 


that  is, 
Hence 


V  =  ■%-  tanh  t  Vkg^ 
^  sinh  t  ^</kg 


*  COShtV/i-gr 


8  =  -  log  cosh  t  y/kg  +  c. 
k 


162.  Transcendental  equations.  Equations  involving  transcen- 
dental functions  can  often  be  solved  by  methods  similar  to  those 
used  for  algebraic  equations.  Graphical  methods  can  often  be  used 
to  advantage. 

Ex.  1.    sin  a;  =  a. 

The  solutions  of  this  equation  are  the  abscissas  of  the  points  of  intersection 
of  the  curve  y  =  sinx  and  the  straight  line  y  =  a  (fig.  167).  If  a  >  1  or  a  <  —  1, 
there  are  no  real  solutions  ;  otherwise  there  are  an  infinite  number  of  solutions. 

TT 

Let  us  call  the  smallest  positive  root  Xj,  where  0  <  xi  <  -  if  a  is  positive,  and 


] 

r 

-Ztt/                            Nc'T                                0 

/                           \7r                              2w/ 

\    y 

\  „.»   /  - 

Fig.  167 

TT  <  Xi  <  2  TT  if  a  is  negative.  The  value  of  Xx  must  be  found  from  a  table  or 
approximately  from  the  graph.  The  next  largest  positive  root  is  then  tt  —  Xi 
when  a  is  positive,  and  3  tt  —  Xi  when  a  is  negative  ;  and  all  other  roots,  positive 
or  negative,  are  found  by  adding  or  subtracting  multiples  of  2  tt.  Hence  the 
general  solution  is  2  A;7r  +  Xi  and  (2  A;  +  1)  rr  -  Xi,  or,  more  compactly  written, 

/c7r  +  (-l)'^^Xi, 

where  k  is  any  positive  or  negative  integer  or  zero. 

Ex.  2.    cosx  =  a. 

The  general  solution  is  2  ^tt  ±  xi,  where  Xi  is  the  smallest  positive  solution 
and  k  is  an  integer  or  zero.    The  proof  is  left  to  the  student. 

Ex.  3.    tanx  =  a. 

The  general  solution  is  kiz  +  Xi.    The  proof  is  left  to  the  student. 


294    ELEMENTARY  TRANSCENDENTAL  FUNCTIONS 


Ex.4,    cos  2  a;  =  2  cos  X. 

When  an  equation  involves  two  or  more  trigonometric  functions  it  is  well  to 
write  it  in  terms  of  one.    The  above  equation  may  be  written 

2  cos^x  —  1=2  cosx, 

which  is  a  quadratic  equation  in  cos  x.    Solving,  we  have  in  the  first  place 

cos  x  =  -J-  ±  -^  V3, 

but  the  plus  sign  may  be  disregarded,  since  for  real  angles  cos  x  is  not  greater 
tlian  1  numerically.    The  equation 

cos  X  =  J  -  1  Vs 

is  now  to  be  solved  as  in  Ex.  2.    There  results  x  =  2  Arir  i  1.946. 

Ex.  5.    tanx  =  kx. 

The  roots  of  this  equation  are  the  abscissas  of  the  points  of  intersection  of 
the  curve  y  —  tanx  and  the  straight  line  y  =  kx  (fig.  168). 


Fig.  168 

The  two  intersect  at  the  origin,  but  the  other  intersections  depend  upon  the 
value  of  k.    Since  the  slope  of  the  curs'e  y  —  tan  x  is  1  when  x  =  0,  and  >  1  when 

0<x<-,  we  need  to  distinguish  three  cases,  according  as  A;  >1,  0<fc  <  1,  or  A;<0. 


TRANSCENDENTAL  EQUATIONS       295 

The  graph  shows  that  if  fc  >  1,  the  smallest  positive  root  lies  between  0  and 
—  ;  if  0  <  A;  <  1,  the  smallest  positive  root  lies  between  tt  and  —  ;  and  if  fc  <  0, 
the  smallest  positive  root  lies  between  —  and  tt. 

We  shall  now  find  the  smallest  positive  root  in  the  special  case 

tan  X  =  2  X. 

We  must  first  locate  the  root  (§  47),  either  by  the  graph  or  by  means  of  a  table. 
If  a  table  is  used,  it  must  be  one  in  which  angles  are  given  in  radians.  We  shall 
use  the  table  on  page  132  of  Professor  B.  O.  Peirce's  "Short  Table  of  Integrals." 
We  find,  by  looking  for  a  place  in  the  tables  where  the  tangent  of  an  angle  is 
approximately  equal  to  twice  the  angle,  that  when  x  =  1.1G36  (G6°  40'),  tanx 
=  2.3183,  and  when  x  =  1. 1665  (66°  50'),  tan  x  =  2.3369.   Consider  now  the  curve 

y  =  tan  x  —  2  x. 

When  xi  =  1.1636,  yi  =  -  .0089,  and  when  Xg  =  1.1665,  yz  =  .0039. 

Hence  the  curve  intersects  OX  between  xi  and  X2,  and  a  root  of  the  equation 

tan  X  —  2  X  =  0 

is  therefore  located  to  two  decimal  places.  To  locate  the  root  more  closely  we 
will  use  the  method  of  §  63.    We  have 

^  =  sec2x  -  2, 
dx 

and  — ^  =  2  tan  x  sec^  x, 

both  of  which  are  positive  when  x  is  between  Xi  and  Xg.  Hence  that  portion  of 
^i^^onrye  y  =  tanx-2x 

appears  as  in  fig.  64,  (1),  and  its  intersection  with  OX  lies  between  the  tangent 
at  (X2, 2/2)  and  the  chord  connecting  {xi,  yi)  and  {X2, 2/2)-    The  tangent  at  (X2, 2/2)  is 

y  -  .0039  =  4.461  (x  -  1. 1665) ; 

0128 
the  chord  is  y  -. 0039  =  '-^—  (x -  1 .  1665) , 

and  the  point  of  intersection  of  each  with  OX  is  found  to  be 

x  =  1.1656 
to  four  places  of  decimals.    This  is  therefore  the  root  of  the  equation  to  four 
decimal  places. 

Ex.6.    e^-4x2-2x  +  3  =  0. 

The  roots  of  this  equation  are  the  abscissas  of  the  points  of  intersections  of 
the  curves  y  =  e^  and  j/  =  4x2  +  2x  -  3,  and  may  be  found  graphically  or  by 
means  of  tables  to  lie  between  -  1  and  -  2  and  between  0  and  1.  To  determine 
the  root  between  0  and  1,  we  place  y  =  e^  -  4x2  -  2x  +  3.  When  Xi  =  0, 
2/1  =  4,  and  when  X2  =  1,  2/2  =  —  •282. 


296    ELEMEXTAKY  TKAXSCENDENTAL  FUNCTIONS 

Also  ^  =  e^-8x-2, 

dx 

and  —  =  e^  -  8, 

which  are  both  negative  when  x  is  between  0  and  1.  Hence  the  portion  of  the 
curve  in  question  has  the  shape  of  fig.  64,  (4),  and  its  intersection  with  OX  lies 
between  that  of  the  tangent  at  {x^,  y^)  and  that  of  the  chord  connecting  (xi,  yi) 
and  (Xj,  yo).    The  tangent  is 

y +  . 282  =  - 7.282  (x-1), 
which  intersects  OX  when  x  =  .97  — .    The  chord  is 

2/ +  .282  =  - 4.282  (x-1), 
which  intersects  OX  when  x  =  .93  +  . 

If  we  now  place  Xi  =  .93,  yi  =  .2149,  and  if  Xj  =  .97,  2/2  =  —  -0657,  the  tan- 
gent  at  (x^,  y,)  is  ^  ^  ^.„  ^  _  ^^^gi  (^  _  .97), 

which  intersects  OX  where  x  =  .9608—  ;  and  the  chord  between  (Xi,  yi)  and 

(«2,  Vi)  is  _  oaof? 

y  +  .0657  =  (X  -  .97), 

.04      ^  ^' 

which  intersects  OX  where  x  =  .9606+. 

Hence  a  root  of  the  equation  lies  between  .9606  and  .9608. 

PROBLEMS 
Plot  the  graphs  of  the  following  equations : 

1.  y  =  ctnx,  11.  y  =  xsin-. 

x 

2.  y  =  secx.  j 

_  12.  y  =  x^sin-- 

6.  y  —  cscx.  x 

4.  y  =  versx.  .  13.  y  =  el^. 

5.  y  =  ^  sin  3  x.  L-^x 

is  ,10  14.  y  =  xe  ^  . 

6.  y  =  sm  X  +  ^  sin  3  X.  ^ 

7.  y  =  sin  X  +  sin  2  x.  15.  y  =  xe^. 

8.  y  =  2  sin  X  -  sin  2  X.  16.  y  =  log  (sin  x). 

9.  y  =  cosx+^cos3x.  17.  2/ =  tan- 1  (ox  +  6). 

X  —  1 

10.  y  =  1  -  A  cosx  -  *  cos 2 x.  18.  y  =  log . 

■*  X  +  1 

19.  Plot  the  graph  of  the  equation  y  =  -  sin  x,  and  determine  what  relation 
it  has  to  the  hyperbolas  xy  =  ±  1. 

20.  Plot  the  graph  of  the  equation  y  =  sin  x',  and  show  that  the  distance 
between  two  consecutive  intercepts  on  OX  approaches  zero  as  a  limit. 


PROBLEMS  297 

dt/ 
Find  —  in  each  of  the  following  cases : 
dx 

21.  2/ =  sin  (ox  +  6)  cos  (ax  —  6).  45.  y  =  sin- 1  (2  x  Vl  —  x^). 

22.  y  =  tan  (ax  +  6)  ctn  (ox  +  c).  ._  ,    2x  — 1 

46.  y  =  csc-i 


23. 

8ec4x 

24. 

ctn  2  X  +  2 

y  — • 

csc2x 

25. 

sec3x 

y  =  - — ^ — -^- 

2  V^ 


X 


47.  y  =  sec- 1  V4  x2  +  4 X  +  2. 

48.  y  =  CSC-  •  -  I  x2  +  —  I . 

49.  y  =  e^  +  2ar. 


lauox-t-i  L2  _  a2 

26.  y  =  csc2x-ctn2x.  50.  y  =  \ogy^^^  ^  ^^- 

27.  y  =  sec"'nxcsc»mx.  ^         

28.  2/  =  sec22x  +  tan2x.  51.  y  =  a^i-=^e^i-^. 

29.  1/=  ctn4xcsc2x.  52.  y  =  log(2x  +  1  +  2  Vx2  +  x). 

30.  y  =  sin(xcosx).  53.  y  =  e»: -v^a. 

31.  y  =  (cos2x  +  f)sin8x.  54.  y  =  o*""^. 

32.  2/  =  (2  sec^x  +  3  sec2x)  sinx.  55.  y  =  x^  logx2. 
sin2x  56.  y  =  otan^^ec^'^. 


33.  y  = 


Vl  —  cos2x  57.  y  =  tan  2  x  •  a"*= '- "'. 


34.  y  =  cos  Vl  —  x2.  58.  y  =  e(<'  +  *)''sinmx. 

__  sin2x      cos2x  59.  w  =  csc-i(sec2x). 

35.  y  = _ 

sinx  COSX  CA  /       ,       V    tan-iV^ 

SO.  y  —  {x  +  a)e        ^  <-. 

36.  y  =  sin-i     :•  ,  e"' -  e-^ 

Vi  +  x2  61-  2'  =  ta"-'^^rqr^x- 

37.  y  =  tan-i-^=.  g2    ,,  -  ip..  ^  ^^" '^  +  ^ 

Vrr^  '*''•  ^~  "^^  tanx  +  3 

38.  y  =  cos-i:l-^-  .     63.  y  =  sec-i^ — 

1  +  x  ^ 

(J  64.  y  =  e^=<'««cos(xsina). 

39.  y  =  sec- 1     — --•  65.  2/  =  log  tan  (x2  +  a2). 


a-x  66.  y  =  x  cos->x  -  Vl  -  x2. 

40.  y  =  sin-i ^ 

"  +  *  67.  2/  =  -  log  (sec  ax  +  tan  ox). 

41.  y  =  sec-il(|Vx  +  ^y  gg    ,^^(a;  +  oVr3^)e«--'-. 

42.  2/  =  CSC- 1  (x2  +  2  X).  69.  y  =  log  Vl  +  x2  +  x  ctn- 1  x. 

43.  2/  =  ctn-i^^-2tan-:l  70.  y  =  ^^  -  log  VT^. 
^               x2  -  a2                 a  Vl-x2 

..  1  ,6  +  acosx  _.  e<^(o  sin  mx  —  m  cos  mx) 

44.  y  =  —  cos-^ ; 71.  v  =  — ^ ; ;, " 

Va2  -  62  a  +  6cosx  "  m^  +  a^ 


298    ELEMENTARY  TRAJSTSCENDENTAL  FUNCTIONS 


72.  y  =  a;2  ctn-i  _  -|-  a^  tan-i ax. 

X  a 

-g  ,      ,  *  X     sin-ix 

X 

X 


y  =  \og( ^ )- 

\1  +  Vl-X2/ 


74.  y  =  log(x  +  Vx2  -  a2)  +  cos-i 

75.  y  =  log(x  +  Vx2  -  a2)  _  csc-i-. 

76.  y  =  tan-i  Vx^  -  2x  -  Ig^jg^lj). 

Vx2  —  2  X 


,/    /a -6^      x\ 

=^  Un- 1(  \ tan  -  I . 

b2  \\a  +  6        2/ 


77.2/  =  -^ 
Va2 

78.  2/  =  log  tan  (2  x  +  1)  +  esc  (4  x  +  2). 


mn  r^ ',  1  "^2  OX  —  X2 

79.  y  =  v2  ax  —  x^  +  a  cos-i 


80.  2/  =  X Va2  +  x2  +  a21og(x  +  VoM^). 


o,  ,  2  v^  -  1      2V^sec-'2Vx 

81.  y  =  log  ^  — 

\2Vx+l  V4x-1 

ort  X  —  a  ,/r r  ,  a*^  .     ,x  — o 

82.  y  = V2ax  — x2-) sin-i 


83.  y  =  tan- 1  (x  -  Vx2  -  1)  +  log 


Vx*  - 1 


84.  y  =  -M^±IL  +  csc-i  V^^^T^. 

Ve2^+  2e*-  1 

85.  y  =  ^  Vl  -  x2  _  A  _  ?!\  tan-i  Vl  _  x2. 

86.  2/  =  -^  log    ,   ^  +  log(x  +  Vl  +  x2). 

2V2        V2  +  2x2  +  x 

87.  y  =  (sin  V^)'*°^.  90.  y  =  (x)««. 

88.  y  =  </^nx.  91-  ^  =  (e)=^. 

89.  y=x(^).  92.  y  =  (o  +  x)*^* (»+*>. 

Find  —  in  each  of  the  following  cases : 
dx 

93.  x!'  +  secxy  =  0.  97.  e^ sin y  -  ei' cos x  =  0. 

94.  ytan-ix-2/2  +  x2  =  0.  ^8.  y sin x  +  x cos 2/ =  x2^. 

95.  ysinx  —  cos  (x  —  y)  =  0, 


99.  7/  logx  =  xsiny. 


ni>  100.  X2/  =  tan-i-. 

96.  ye"v  =  ox™.  '^  y 


PROBLEMS  299 


1-x     ,     1+x 


„.    ,  dy    d'hj    dhj 

Find  -^,  -4,  -4  m  each  of  the  following 
dz    dx^    dx^  ^ 

101.  log  (x2  +  2/2)  -  2  tan-i  ?^  =  0.  103.  log  ^— ^  -  log  i^t^  =  1. 

X  l+y  l-y 

102.  e-  +  ev  =  l.  104.  x-2/  =  log(a;  +  y). 

105,  &^+y  =  x«. 

106.  At  what  points  is  the  curve  j/  =  sin  x  +  sin  2  x  parallel  to  the  axis  of  x  ? 

107.  What  value  must  be  assigned  to  m  that  the  curve  y  = h  tan-'(x+m) 

bmx 
may  be  parallel  to  OX  at  the  point  the  abscissa  of  which  is  1  ? 

108.  Find  the  angle  of  intersection  of  the  curves  3/  =  sinx  and  y  =  cosx. 

109.  Find  the  angle  of  intersection  of  the  curves  y  =  sin  x  and  y  =  sin  (x  +  a). 

110.  Find  the  angle  of  intersection  of  the  curves  y  =  sinx  and  y  =  sin2x. 

111.  Show  that  the  portion  of  the  tangent  to  the  curve 
o,     a  +  Va2  -  x2 


y  =  -  log ===:  —  V  a^  —  x2 

2       a  -  Va2  -  x2 

included  between  the  point  of  contact  and  the  axis  of  y  is  constant.    (From  this 
property  the  curve  is  called  the  tractrix.) 

112.  Find  the  points  of  inflection  of  the  curve  y  =  2  sin  x  —  ^  sin  2  x. 

113.  Find  the  points  of  inflection  of  the  curve  xy  =  a^log-- 

114.  Find  the  points  of  inflection  of  the  curve  y  =  e-^ 

115.  Prove  that  the  curve 

y  =  ^x  —  ^sinx  +  j^  sin  2  x 

has  an  indefinite  number  of  points  of  inflection,  and  that  two  of  them  lie  between 
the  points  for  which  x  =  6  and  x  =  10  respectively. 

116.  Plot  the  curve  y  =  sin^x,  finding  maxima  and  minima,  and  points  of 
inflection. 

117.  Plot  the  curve  y  =  e-«^cos6z,  and  prove  that  it  is  tangent  to  the  curve 
y  =  g-oa:  wherever  they  have  a  point  in  common.  Find  maxima  and  minima  and 
points  of  inflection  of  this  curve  when  a  =  6  =  1. 

118.  Plot  the  curve  y  =  x"e-^  (n  >  0),  finding  maxima  and  minima  and  points 
of  inflection. 

119.  A  body  moves  in  a  plane  so  that  x  =  a  cost  +  b,  y  =  a sint  +  c,  where 
t  denotes  time  and  a,  6,  and  c  are  constants.  Find  the  path  of  the  body,  and 
show  that  its  velocity  is  constant. 

120.  A  rectilinear  motion  is  expressed  by  the  equation  s  =  5  —  2  cos' t.  Show 
that  the  motion  is  a  simple  harmonic  motion,  and  express  the  velocity  and  the 
acceleration  at  any  point  in  terms  of  s. 


300    ELEMENTARY  TRANSCEKDE:N^TAL  FUNCTIOXS 

121.  A,  the  center  of  one  circle,  is  on  a  second  circle  witli  center  at  B. 
A  moving  straiglit  line,  AMN,  intersecting  the  two  circles  at  M  and  N  respec- 
tively, has  constant  angular  velocity  about  A.  Prove  that  BN  has  constant 
angular  velocity  about  B. 

122.  Two  particles  are  moving  on  the  same  straight  line,  and  their  dis- 
tances from  the  fixed  point  O  on  the  line   at  any  time  t  are   respectively 

X  =  a  cos  ut  and  x'  =  a  cos  lut-] —  j,  w  and  a  being  constants.  Find  the  greatest 
distance  between  them.      ^  ' 

123.  A  ladder  b  ft.  long  leans  against  a  side  of  a  house.  Its  foot  is  drawn 
away  in  the  horizontal  direction  at  the  rate  of  a  ft.  per  second.  How  fast  is  its 
center  moving  ? 

124.  If  a  particle  moves  so  that 

s  =  e-  2  «^'  (a  sin  ht  +  b  cos  hi), 

find  expressions  for  the  velocity  and  the  acceleration.  Hence  show  that  the 
particle  is  acted  on  by  two  forces,  one  proportional  to  the  distance  from  the  origin 
and  the  other  proportional  to  the  velocity.    Describe  the  motion  of  the  particle. 

125.  If  s  =  ae*"'  -|-  be-'-',  show  that  the  particle  is  acted  on  by  a  repulsive 
force  which  is  proportional  to  the  distance  from  the  point  from  which  s  is 
measured. 

126.  BC  is  a  rod  a  ft.  long,  connected  with  a  piston  rod  at  C,  and  at  B  with 
a  crank  AB  b  ft.  long,  revolving  about  A.  Find  Cs  velocity  in  terms  of  AB's 
angular  velocity. 

127.  A  man  walks  along  the  diameter,  200  ft,  in  length,  of  a  semicircular 
courtyai-d  at  a  uniform  rate  of  5  ft.  per  second.  How  fast  will  his  shadow  move 
along  the  wall  when  the  rays  of  the  sun  are  at  right  angles  to  the  diameter  ?    . 

128.  How  fast  is  the  shadow  in  the  preceding  problem  moving  if  the  sun's 
rays  make  an  angle  a  with  the  diameter  ? 

129.  Given  that  two  sides  and  the  included  angle  of  a  triangle  have  at  a 
certain  moment  the  values  6  ft.,  10  ft.,  and  30°  respectively,  and  that  these 
quantities  are  changing  at  the  rates  of  3  ft.,  —  2  ft.,  and  10°  per  second  respec- 
tively, what  is  the  area  of  the  triangle  at  the  given  moment,  and  how  fast  is  it 
changing  ? 

130.  One  side  of  a  triangle  is  I  ft.,  and  the  opposite  angle  is  a.  Find  the 
other  angles  of  the  triangle  when  its  area  is  a  maximum. 

131.  A  tablet  8  ft.  high  is  placed  on  a  wall  .so  that  the  bottom  of  the  tablet 
is  20  ft.  from  the  ground.  How  far  from  the  wall  should  a  person  stand  in  order 
that  he  may  see  the  tablet  at  the  best  advantage,  i.e.  in  order  that  the  angle 
between  the  lines  from  the  observer's  standpoint  to  the  top  and  the  bottom  of 
the  tablet  may  be  the  greatest  ? 

132.  A  weight  P  is  dragged  along  the  ground  by  a  force  F.  If  the  coefficient 
of  friction  is  K,  in  what  direction  should  the  force  be  applied  to  produce  the 
best  result  ? 


PROBLEMS  301 

133.  An  open  gutter  is  to  be  constructed  of  boards  in  such  a  way  that  the 
bottom  and  the  sides,  measured  on  the  inside,  are  to  be  each  4  in.  wide,  and 
both  sides  are  to  have  the  same  slope.  How  wide  should  the  gutter  be  across 
the  top  in  order  that  its  capacity  may  be  as  great  as  possible  ? 

134.  Above  the  center  of  a  round  table  is  a  hanging  lamp.  What  must  be 
the  ratio  of  the  height  of  the  lamp  above  the  table  to  the  radius  of  the  table 
that  the  edge  of  the  table  may  be  most  brilliantly  lighted,  given  that  the  illumi- 
nation varies  invereely  as  the  square  of  the  distance  and  directly  as  the  cosine 
of  the  angle  of  incidence  '? 

135.  A  steel  girder  27  ft.  long  is  to  be  moved  on  rollers  along  a  passageway 
and  into  a  corridor  8  ft.  in  width  at  right  angles  to  the  passageway.  If  the  hori- 
zontal width  of  the  girder  is  neglected,  how  wide  must  the  passageway  be  in 
order  that  the  girder  may  go  around  the  corner  ? 

136.  Find  the  area  of  an  arch  of  the  curve  tj  =  sin  x. 

137.  Find  the  area  bounded  by  the  axis  of  y  and  the  portion  of  the  curves 
?/  =  sin  X,  y  =  cos  x,  lying  between  x  =  0  and  x  =  tt. 

138.  Find  the  area  bounded  by  the  portions  of  the  curves  y  =:  ^  sin  2  x  and 
?/  =  sin  X  +  ^  sin  2  x  that  extend  between  x  =^  0  and  x  —  tt. 

139.  Find  the  area  between  the  curve  y  =  e^,  the  axis  of  x,  and  the  ordinates 
X  =  0  and  x  =  1. 

140.  Find  the  area  bounded  by  the  axis  of  x,  the  catenary,  and  the  ordinates 
X  =  ±  a. 

141.  Find  the  area  bounded  by  the  axis  of  x,  the  curve  y  =  -,  and  the 
ordinates  x  =  1  and  x  =  2.  ^ 

142.  Find  where  the  ordinate  of  the  witch  should  be  drawn  in  order  that 
the  area  between  that  ordinate,  the  witch,  the  axis  of  y  and  the  axis  of  x  should 
be  equal  to  the  area  of  the  circle  used  in  the  definition. 

143.  Show  that  for  the  catenary  —  =  -(e"  +  e~«),  and  thence  find  an 
expression  for  the  length  of  s. 

144.  Find  the  curve  the  slope  of  which  at  any  point  is  k  times  the  reciprocal 
of  the  abscissa  of  the  point,  and  which  passes  through  (2,  3). 

145..  Find  the  curve  the  slope  of  which  at  any  point  is  k  times  the  ordinate 
of  the  point,  and  which  passes  through  the  point  {a,  h). 

146.  Find  the  space  traversed  by  a  moving  body  in  the  time  t  if  its  velocity 
is  proportional  to  the  distance  traveled. 

Solve  the  following  equations : 

147.  tanx  =  cosx.  152.  tanx  =  x. 

148.  cos2x  =  Jcosx.  153.  tanx=^x. 

149.  sin  2  tf  cos  2  fl  4-  2  sin  ^  =  0.  154.  x  -  ^  .sin  x  =  ■^^. 

150.  sin4x  -  2sinxcos2x  =  0.  155.  e^=«2. 

151.  sin*x+3cos*x-4sin2xcos2x  =  0.  156.  logx  =  ^x. 


CHAPTEE  XIV 

PARAMETRIC  REPRESENTATION  OF  CURVES 

163.  Definition.  Thus  far  we  have  considered  a  curve  as 
represented  by  a  single  equation  connecting  x  and  y.  Another 
useful  method  is  to  express  x  and  y  each  as  a  function  of  a 
third  independent  variable  ;  thus : 

where  t  is  an  independent  variable  and  f^{f)  and  f^{t)  are 
continuous  functions  of  t.  As  t  varies,  x  and  y  also  vary, 
and  the  point  {x,  y)  traces  out  a  curve.  By  eliminating  t  be- 
tween  the    two    equations    the    curve    may   often    be    expressed 

by  a  single  equation  between 
X  and  y. 

164.  The  straight  line. 
Let  Pi{x^,  2/1)  (fig-  169)  be  a 
fixed  point  on  a  straight  line 
and  <^  be  the  angle  which 
the  line  makes  with  a  line 
P^R  parallel  to  OX.  Let 
P{x,  y)  be  any  point  on 
the  line,  and  r  the  distance 
from  P^  to  P,  where  r  is 
positive  when  P  is  on  the 
terminal  line  of  ^,  and  negative  when  P^  is  on  the  backward  exten- 
sion of  the  terminal  line.    Then,  for  all  possible  positions  of  P 


Fig.  169 


x  —  x^  y  —  y^ 

— -— =  cos</),         — -^  =  sin<^; 


whence  x  =  x^-\-r  cos ^,         y  =  y^-\-r  sin <^. 

302 


THE  CIRCLE  AND  THE  ELLIPSE 


303 


This  is  a  parametric  representation  of  the  straight  line,  where 
r  is  the  arbitrary  parameter.  Illustrations  of  the  use  of  these 
equations  have  been  given  in  §§136  and  138. 

Another  parametric  representation  of  a  straight  line  is  furnished 
by  the  equations  of  §  19, 

X  =  X^  -f-  fr  \^^  ~~  '^i/i 

where  I  is  the  parameter  and  {x^,  y^  and  {x^,  y^  are  fixed  points. 
More  generally,  the  equations 

x  =  a-\-ht,  y=zf  +  gt, 

where  a,  b,  f,  g,  are  constants,  and  t  is  an  arbitrary  parameter, 
represent  a  straight  line.    For  these  equations  are  equivalent  to 


165.  The    circle.      Let 

P{x,  y)  (fig.  170)  be  any 
point  on  a  circle  with  its 
center  at  the  origin  0,  and 
its  radius  equal  to  a.  Let  ^ 
be  the  angle  made  by  OP 
and  OX.  Then  from  the  defi- 
nition of  the  sine  and  cosine 

x  =  a  cos<f>, 
y  —  d  sin^, 


Fig.  170 


the  parametric  equations  of  the  circle  with  <^  as  the  arbitrary 
parameter. 

166.  The  ellipse.    Take  the  ellipse 


^  +  ^  =  1 


(a,>h) 


and  on  its  major  axis  as  a  diameter  construct  a  circle.    Take 
F{x,  y)  (fig.  171)  any  point  on  the  ellipse,  draw  the  ordinate  MF 


304    PARAMETRIC  REPRESENTATION  OF  CURVES 


and  prolong  it  imtil  it  meets  the  circle  in  Q.    Call  the  coordinates 
of  Q  {x,  y').    Then  from  the  equation  of  the  circle 


y 


'=V^^ 


and   from  the  equation  of 
the  ellipse 


Fig.  171 


^      a 
^   Hence  y  ==  —  y'. 

Draw  the  line  OQ,  mak- 
ing the  angle  XOQ  =  <f). 
Then,  as  in  §  165, 

x  =  a  cos  <^, 
y'  =  a  sin  cf). 


By  substituting  for  y'  its  value  in  terms  of  y,  we  have 

X  —  a  cos  (f>,  y  =  ^  sii^  ^> 

the  parametric  equations  of  the  ellipse. 

^  is  called  the  eccentric  angle  of  a  point  on  the  ellipse,  and 
the  circle  x^+'i^=o?'  is  called  the  auxiliary  circle. 

Ex.  The  parametric  equations  of  an  ellipse  may  be  used  to  find  its  area. 
For  if  A  is  the  area  bounded  by  the  ellipse,  the  axis  of  y,  the  axis  of  x,  and 
any  ordinate  MP  (fig.  171),  then  (6,  §  109) 

dA 

—  =  V- 
dx 

dA  dA 

dA  _d^_        d<t> 
dx       dx       — asinc^ 


But 


(1) 


and  y  =  6  sin  <^, 

Therefore  (1)  is  equivalent  to 
dA 
d(j) 

Hence  A 


=  abf 


ah  sin2  (p  =  ab 
sin  2  0      <f> 


cos  2  (^  —  1 


+  c. 


THE  CYCLOID 


305 


When  <6  =  - ,  ^  =  0 ;  hence  c  = . 

2'  4 


Therefore 


A  =  ab(^^ 


2       4/ 


When  (f>  =  0,  A  is  one  fourth  the  area  of  the  ellipse.  Therefore  the  whole 
area  of  the  ellipse  equals  Trab. 

167.  The  cycloid.  If  a  circle  rolls  upon  a  straight  line  each 
point  of  the  circumference  describes  a  curve  called  a  cycloid. 

Let  a  circle  of  radius  a  roll  upon  the  axis  of  x  and  let  C 
(fig.  172)  be  its  center  at  any  time  of  its  motion,  N  its  point  of 


contact  with  OX,  and  P  the  point  on  its  circumference  which 

describes  the  cycloid.    Take  as  the  origin  of  coordinates,  0,  the 

point  found  by  rolling  the  circle  to  the  left  until  P  meets  OX. 

Then 

OJ:i=  arc  FK 

Draw  3fP  and  CJSF  each  perpendicular  to  OX,  PR  parallel  to 
OX,  and  connect  C  and  P.    Let 

NOP  =  0. 

Then  x  =  OM=ON-  MN 

=  arc  NP  -  PR 
=  a(j)  —  a  siu  </>. 
y  =  MP  =  NC  —  RC 
=  a  —  a  cos  (f). 

Hence  the  parametric  representation  of  the  cycloid  is 

x=  a(<j)  —  sin  <f)), 
y  =  a(l—  cos<f>). 


306     PARAMETRIC  REPRESENTATION  OF  CURVES 
By  eliminating  <j>  the  equation  of  the  cycloid  may  be  written 


x==  a  cos 


but  this  is  less  convenient  than  the  parametric  representation. 

At  each  point  where  the  cycloid  meets  OX  a  sharp  vertex  called 
a  cusp  is  formed.  The  distance  between  two  consecutive  cusps  is 
evidently  2  rra. 

168.  The  trochoid.  When  a  circle  rolls  upon  a  straight  Une, 
any  poiat  upon  a  radius,  or  upon  a  radius  produced,  describes  a 
curve  called  a  trochoid. 


M    ^ 


Fig.  173 


Let  the  circle  roll  upon  the  axis  of  x,  and  let  C  (figs,  173  and 
174)  be  its  center  at  any  time,  N  its  point  of  contact  with  the 
axis  of  X,  F(x,  y)  the  point  which  describes  the  trochoid,  and 


Fig.  174 


K  the  point  in  which  the  liae  CP  meets  the  circle.  Take  as 
the  origin  0  the  point  found  by  rolHng  the  circle  toward  the 
left  until  K  is  on  the  axis  of  x.    Then 

ON^iixcNK. 


THE  EPICYCLOID 


307 


Draw  PM  and  CN  perpendicular  to  OX,  and  through  P  a  line 
parallel  to  OX,  meeting  CN  or  CN  produced,  in  R.  Let  the  radius 
of  the  circle  be  a,  CP  be  h,  and  NCP  be  <^.    Then 

x=^OM  =  ON-MN 

=  sivc  NK-PB 

=  a^  —  h  sin  <f). 
y  =  MP  =  NC-BC 

=  a  —  h  cos  <^. 

169.  The  epicycloid.  When  a  circle  rolls  upon  the  outside  of 
a  fixed  circle,  each  point  of  the  circumference  of  the  rolling  circle 
describes  a  curve  called  an  epicycloid. 

r 


Fig. 175 


Let  0  (fig.  175)  be  the  center  of  the  fixed  circle,  C  the  center  of 
the  rolling  circle,  N  its  point  of  contact  with  the  fixed  circle,  and 


308    PAKAMETRIC  REPRESENTATION  OF  CURVES 

P{x,  y)  the  point  which  describes  the  epicycloid.    Determine  the 

point  K  by  rolling  the  circle  C  until  P  meets  the  circumference 

of  0.    Then 

arc  KN  =  arc  NP. 


Take  0  as  the  origin  of  coordinates,  and  OK  as  the  axis  of  x. 
Draw  PM  and  CL  perpendicular  to  OX,  PS  parallel  to  OX,  meet- 
ing CL  in  R,  and  connect  O  and  C.  Let  the  radius  of  the  rolling 
circle  be  a,  that  of  the  fixed  circle  h,  and  denote  the  angle  OOP 
by  e,  the  angle  KOC  by  4>.    Then 


whence 


arc  KN  =  h(f),         arc  NP  =  aO : 


We  now  have      x  =  Oil  =  OL-h  LM 

=  OC  cos  KOC  -  CP  cos  SPC 

=  (a  -i-h)  cos  <f>  —  a  cos  (^  +  ^) 

=  (a  +  6)  cos  9  —  a  cos 9. 

y  =  MP  =  LC-RC 
=  OC  sin  KOC  -  CP  sin  ^PC 

=  (a  +  &)  sin  ^  —  a  sin  (^  +  ^) 

/     ,   7,\    •    ji           .    a  +  b 
=  (a  +  0)  sm  9  —  a  sin 9. 


The  curve  consists  of  a  number  of  congruent  arches  the  first  of 
which  corresponds  to  values  of  0  between  0  and  2  ir,  tliat  is,  to- 

2(177 


values  of  ^  between  0  and 


Similarly  the  Xth  arch  corre- 


sponds to  values  of  <j>  between  — — -— ^ —   and   — Hence 

0  b 

the  curve  is  a  closed  curve  when,  and  only  when,  for  some  value 

of  k,  ^^ —  is  a  multiple  of  2  tt.    If  a  and  b  are  incommensurable, 
0 

CL         7)  10 

this  is  impossible,  but  if  -  =  —  >  where  —  is  a  rational  fraction  in 

b      q  q 

its  lowest  terms,  the  smallest  value  of  Tc  =  q.    The  curve  then  con- 
sists of  q  arches  and  wiuds  p  times  around  the  fixed  circle. 


THE  HYPOCYCLOID 


309 


170.  The  hypocycloid.  When  a  circle  rolls  upon  the  inside  of 
a  fixed  circle,  each  point  of  the  rolling  circle  describes  a  curve 
called  the  hypocycloid.  If  the  axes  and  the  notation  are  as  in 
the  previous  article,  the  equations  of  the  hypocycloid  are 

x  =  (b  —  a)  cos  ^-\-  a  cos rf>, 

a 

y  =  (b  —  a) sin <f)—  a  sin (f>. 

The  proof  is  left  to  the  student.    The  curve  is  shown  in  fig.  176. 


Fig. 176 


171.  Epitrochoid  and  hypotrochoid.  The  epitrochoid  and 
hypotrochoid  are  generated  by  the  motion  of  any  point  on 
the   radius   of   a    circle    which   rolls    upon   the    outside  or   the 


310    PARAMETRIC  REPRESENTATION  OF  CURVES 

inside  of  a  fixed  circle.  If  h  is  the  distance  of  the  generating 
point  from  the  center  of  the  moving  circle,  and  the  notation 
is  otherwise  the  same  as  in  the  previous  articles,  the  equations 
of  the  epitrochoid  are 

x  =  {a-\-h)  cos  (f>  —  h  cos (f>, 

y  =  (a  +  b)  sin.(f)  —  h  sm c^i, 

a 


Fig.  177 


and  of  the  hypotrochoid  are 


T    

x  =  (b~  a)  cos  <f)  +  h  cos (f>, 

y  =  (h  —  a)  sin  d>  —  h  sin 6. 

a 


THE  EPITROCHOID 


311 


The  proofs  are  left  to  the  student.    The  curves  are  shown  in 
figs.  177,  178,  and  179,  180  respectively. 


Fig. 178 


172.  The  involute  of  the  circle.  If  a  string,  kept  taut,  is 
unwound  from  the  circumference  of  a  circle,  its  extremity 
describes  a  curve  called  the  involute  of  the  circle.  Let  0  (fig.  181), 
be  the  center  of  the  circle,  a  its  radius,  and  A  the  point  at  which 
the  extremity  of  the  string  is  on  the  circle.  Take  0  as  the  origin 
of  coordinates  and  OA  as  the  axis  of  x.  Let  P  {x,  y)  be  a  point  on 
the  involute,  PK  the  line  drawn  from  P  tangent  to  the  circle  at 
K,  and  ^  the  angle  XOK.  Then  PK  represents  a  portion  of  the 
unwinding  string,  and  hence 

KP  =  Q.VC  AK  =  a<^. 


THE  INVOLUTE  OF  THE  CIRCLE 


313 


Now  it  is  clear  that  for  all  positions  of  the  point  K,  OK  makes 

TT 

an  angle  </>  —  -^  with  0  Y.    Hence  the  projection  of  OK  on  OX  is 
always  OK  cos^=  a  cos^,  and  its  projection  on  OY  is  OK  cos 

7r\  .  TT 

^  —  —  I  =  «  sin<^.    Also  KP  always  makes  an  angle  <f>~  ^  with 


( 


Fig.  181 


OX  and  ir  —  (j>  with  0  Y.    Hence  the  projection  of  KP  on  OX 


is  KP  cos(<^  — —  I  =  a<^  sin^,  and  its  projection  on  OF  is  KP 

cos  (tt  —  (/>)  =  —  a^  cos<f>.    The  projection  of  OP  on  OX  is  x,  and 
upon  OY  is  2/.    Hence,  by  the  law  of  projections,  §  15, 

X  =  a  cos^  +  a<f>  sin(f>, 
y  =  a  sin  ^  —  ci<^  cos  ^. 

173.  Time  as  the  arbitrary  parameter.  An  important  use  of  the 
parametric  representation  of  curves  occurs  in  mechanics  in  find- 
ing the  path  of  a  moving  point  acted  on  by  known  forces.  Here 
the  independent  parameter  is  usually  the  time. 


314    PARAMETRIC  REPRESENTATION  OF  CURVES 

Ex.  1.  A  particle  moves  "in  a  circle  with  uniform  velocity,  k.  Then,  if  s 
represents  the  arc  traversed,  and  a  the  radius  of  the  circle, 

s  =  kt    and    d>  =  -  =  — . 
a      a 

Therefore  the  equations  of  the  circle  are  (§  165), 

U 
x  =  a  cos  — , 
a 

.   kt 
w  =  a  sin  — . 
a 

This  shows  that  the  projections  of  P  on  the  coordinate  axes  have  simple 
harmonic  motions  of  the  same  amplitude. 

Ex.  2.  A  particle  Q  moves  with  uniform  velocity  along  the  auxiliary  circle 
of  an  ellipse  (§  166) ;  required  the  motion  of  its  accompanying  point,  P. 

kt 
As  in  Ex.  1,  0  =  — .    Hence  the  equations  of  the  path  are 

kt 
X  =  a  cos  — , 
a 

^  .    kt 
y  =  0  sm  — , 
a 

showing  that  the  projections  of  P  upon  OX  and  OY  have  simple  harmonic 
motion  of  amplitudes  a  and  b  respectively. 

Ex.  3.  A  projectile  is  shot  with  an  initial  velocity  vq  in  an  initial  direction 
which  makes  an  angle  a  with  the  horizontal  direction.  Then  the  initial  com- 
ponent of  velocity  in  the  horizontal  direction  is  vo  cos  a  and  in  the  vertical 
direction  is  Vq  sin  a.  If  the  resistance  of  the  air  is  neglected,  the  only  force 
acting  on  the  projectile  is  that  of  gravity.  Hence  if  we  take  the  origin  at  the 
initial  position  of  the  projectile,  and  the  axis  of  x  horizontal,  we  have 


¥'="' 

d^y  _ 
dt^          ^' 

which  give 

X  =  Cit  -r  C2, 

But  when  t  = 

=  0,  we  have 
X  =  0, 

y  = 

=  0. 

y  =  -yt^  +  Cst  +  €4. 

dx                        ,  dy 

,  —  =  Vo  cos  a,  and  -^  : 
dt                             dt 

Vo  sm  a. 

Hence  the  parametric  equations  of  the  path  of  the  projectile  are 

X  =  Vot  cos  a, 
y  —  Vot  sin  or  —  J  gt^. 

Eliminating  t  from  these  equations,  we  have 

y  =  X  tan  a  — 


2  Vq  cos2  a 

or  2  Vq  y  cos^  a  =  2  Vo^x  sin  a  cos  a  —  gx^ 

which  shows  that  the  curve  is  a  parabola. 


THE  DERIVATIVES  315 

174.  The  derivatives.   When  a  curve  is  defined  by  the  equations 

dy 

(1) 

dt 
Ex.  For  the  cycloid  q 

X  =  a{<(>  —  sin^), 


we  have,  by  (8).  §96.  |  =  | 


y  =  a(l  —  cos^y, 

dy 

dy      d4>          a  sin  <t> 

-cot*^ 

dx      dx      a(l— COS0) 

2 

d<p 

Now  -p  is  the  tangent  of  the  angle 

made  by  the  tangent  with  the  axis  of  x. 

Therefore  this  angle  is • 

*  2       2 

From  this  follows  a  simple  construction  of  the  tangent  and  normal.    For  if 
the  line  NC  (fig.  182)  is  prolonged  until  it  cuts  the  circle  in  Q,  and  PQ  and  PN 

are  drawn,  the  angle  CQP  —  -  •    Hence  PQ  makes  the  angle with  OX 

and  is  therefore  the  tangent.    Pl>i^  being  perpendicular  to  PQ,  is  the  normal. 

If  it  is  required  to  find  -r-^'  we  may  proceed  as  follows : 


d  (dy 
d?y  d  {dy\  _  dt  \dx 
da?      dx  \dx/  dx 

di 


(2) 


Ex.  For  the  cycloid 


dy  _ 

dx~ 

cot-, 
2 

dx 
d(p  ~ 

a(l  —  cos(^)  = 

2  0  sin2  - 
2 

d-^y 

—  cosec2  - 
2             2 

2asin2- 

2 

1 

dx2 

4  a  sin* 

2 

31G    PARAMETRIC  REPRESENTATION  OF  CURVES 


Formula  (2)  may  be  expanded  as  follows: 


d  (dy 


d^y  dx      dj^x  dy 
~dt~~d¥'di 


dxV 
dt) 


dy\_d  Idt  \  _  df 
dt  \dx)      dt\dx\ 

\dil 

^y  dx  dj^x  dy 
cPy  'de'di~~dell 
dj?  /dxV 

\dt} 

/dsV 
By  multiplying  equation  (3),  §  105,  by  ( — )  >  we  have 

\dt/ 

dsY_/dxY     /dyV 
dt)  ~  \dt/      \dt 


(3) 


(4) 


175.  Application  to  locus  problems.  In  finding  the  Cartesian 
equation  of  the  locus  of  a  point  which  satisfies  a  given  condition, 
it  is  often  convenient  to  employ  the  principles  of  parametric  rep- 
resentation ;  for  by  fixing  the  attention  upon  a  single  point  of 
the  required  locus,  it  is  frequently  possible  to  express  its  coordi- 
nates in  terms  of  a  single  parameter.  The  required  equation  is 
then  found  by  eliminating  the  parameter. 

Ex.  1.  Locus  of  the  point  of  inter- 
section of  perpendicular  tangents  to  a 
parabola. 

Let  the  parabola  be  y^  =  i  px 
(fig.  183),  and  let  the  equation  of 
any  tangent  to  it  be  written  (§  88) 


y  =  mx  + 


P 


(1) 


If  m  is  replaced  by 

110 

y=( )x  +  _L. 

\     mj  1 


,  we  have 


Fig.  183 


X 

mm, 

m 


(2) 


as  the  equation  of  a  tangent  perpendicular  to  (1).    Therefore,  if  P(x,  y)  is  the 
point  of  intersection  of  (1)  and  (2),  P  is  any  point  of  the  locus. 


LOCUS  PROBLEMS 


317 


Solving  (1)  and  (2),  we  find 


and 


X  =  —  p 


(S) 
(4) 


which  are  the  parametric  representations  of  the  locus,  the  parameter  evidently 
being  m.  But  for  all  points  of  the  locus  x  =  —  p,  and  (3)  is  the  Cartesian  equation 
of  the  locus.  It  is  to  be  noted  that  in  this  example  the  elimination  of  the  param- 
eter is  unnecessary,  since  one  of  the  equations  does  not  contain  it. 

As  (3)  is  the  equation  of  the  directrix,  we  have  the  proposition :  Perpendicular 
tangents  to  a  parabola  meet  on  the  directrix. 


Ex.  2.  Locus  of  the  point  of 
intersection  of  perpendicular  tan- 
gents to  an  ellipse. 


Let  the  ellipse  be 1 — 

(fig.  184),  and  let  the  equation  of 
any  tangent  to  it  be  written  (§  88) 


y  =  mx  ±  Va^m'^  +  6^. 


Then   the   equation  of   a   tangent 
perpendicular  to  (1)  will  be 


y  =  --± 

rn 


Ia2_ 
\m2 


+  62, 


and  P{x,  y),  the  point  of  inter- 
section of  (1)  and  (2),  will  be  any 
point  of  the  locus. 

Solving  (1)  and  (2),  we  find 


Fig.  184 


±  m  r-v/^  +  62  _  Va2m2  +  62! 


y  = 


»n2  +  l 

\m^ 

-t 

+  62  +  Va2»i- 

+  62 

m2-M 


(8) 


(4) 


as  the  parametric  representations  of  the  locus  in  terms  of  the  parameter  m. 

To  eliminate  m,  we  square  the  respective  values  of  x  and  y  and  add,  the 
result  being 

x2  +  2/2  =  a^  +  62.  (5) 

The  locus  is  seen  to  be  a  circle  concentric  with  the  ellipse  and  having  its  radius 
equal  to  the  chord  joining  the  ends  of  the  major  and  the  minor  axes  of  the  ellipse. 
While  (3)  and  (4)  form  the  explicit  parametric  representation  of  the  locus,  x 
and  y  being  expressed  explicitly  in  terms  of  the  parameter  m,  <{1)  and  (2)  may 
be  regarded  as  the  implicit  parametric  representation  of  the  locus,  for  x  and  y, 
the  coordinates  of  any  point  of  the  locus,  are  expressed  implicitly  in  terms  of  m. 


318    PARAMETRIC  REPRESENTATION  OF  CURVES 


From  this  point  of  view  it  is  evident  that  we  may  eliminate  m  directly  from 
(1)  and  (2)  to  find  the  Cartesian  equation  of  the  locus.  Accordingly  we  write 
(1)  and  (2)  in  the  forms 


y  —  mx  =  ±  Va%i2  +  62, 
my  +  x  —  ±  Va2  +  b^m^, 

and  square  and  add,  the  result  being 

(1  +  m2)  (X2  +  2/2)  =  (1  +  ^2)  (o2  +  62)^ 

or  (1  +  m2)  (x2  +  ^2  _  a2  _  52)  =  0. 

As  1  +  m'  cannot  be  zero,  since  by  hypothesis  m  must  be  real,  we  may 
cancel  out  this  factor.     The  result, 

x2  +  2/2  _  a2  _  62  =  0, 

is  the  same  as  that  found  by  the  previous  method. 

Ex.  3.    Lociis  of  the  foot  of  the  perpendicular  from  the  focus  of  a  parabola  to 
any  tangent. 

Let  the  parabola  be  2/2  =  4px  (fig.  185),  and  let 

P 


y  =  mx  + 


(1) 


be  any  tangent.    Then  the  perpendic- 
ular to  the  tangent  from  the  focus  is 

y  =  -l^{x-p).         (2) 

Their  point  of  intersection,  P(x,  y), 
is  any  point  of  the  locus. 
Solving  (1)  and  (2),  we  find 


Fig. 185 


and 


x  =  0 


y 


(8) 
(4) 


The  locus  is  therefore  x  =  0,  the  tangent  at  the  vertex  of  the  parabola. 

If  we  proceed  from  the  implicit  parametric  representation,  we  may  elimi- 
nate the  parameter  m  by  substituting  in  (1)  its  value  found  from  ('2).  The 
result  is  x[2/2  +  (p  —  x)2]  =  0,  which  breaks  up  into  two  equations,  i.e. 
X  =  0,  y^  +  (x  —  p)2  =0.  As  the  last  equation  represents  a  single  pointj  it  is 
evident  by  the  geometry  of  the  problem  that  the  required  equation  is  x  =  0, 
as  was  found  by  the  other  method. 

We  see  then  that  when  we  eliminate  the  parameter  from  the  equations 
expressing  x  and  y  in  terms  of  it,  we  must  examine  our  result  carefully  to 
be  sure  that  no  extraneous  factor  is  left  in  it. 


LOCUS  PKOBLEMS 


319 


Ex.  4.    Locus  of  the  foot  of  the  perpendicular  from  the  vertex  of  a  parabola 
to  any  tangent. 

Let  the  parabola  be  y'^  =  'ipx  (fig.  186),  and 


y  =  7nx  + 


(1) 


be  any  tangent.    Then  the  perpen- 
dicular to  (1)  from  the  vertex  is 


y  = X. 

m 


Solving  (1)  and  (2),  we  find 
-P 


y  = 


m2  +  l 
P 


(2) 

(3) 

(4) 


m  (m2  +  1) 
as  the  explicit  parametric  represen- 
tation of  the  locus.    The  Cartesian 
equation  of  the  locus  is  most  readily- 
found  by  substituting  in  (1)  the  value  of  m  from  (2),  and  reducing.    The  result  is 

x8 


Fig.  186 


y^  =  - 


(6) 


p  +  x 
which  is  the  equation  of  a  cissoid  (§  83)  situated  on  the  negative  axis  of  x. 

The  last  two  loci  are  special  examples  of  pedal  curves,  i.e.  loci  of  the  feet  of 
perpendiculars  drawn  from  any  chosen  fixed  point  to  tangents  to  a  given  curve. 

176.  In  the  examples  of  the  last  article  the  parametric  repre- 
sentation of  the  locus  was  in  terms  of  a  single  parameter.  In 
the  examples  of  this  article  the  parametric  representation,  whether 
implicit  or  explicit,  is  in  terms  of  two  parameters,  which  are  not 
independent,  however,  since  they  are  connected  by  a  single  equa- 
tion. The  problem  of  finding  the  Cartesian  equation  of  the  locus 
is,  then,  the  elimination  of  two  parameters  from  three  equations. 

Ex.  1.  Through  the  vertex  of  a  parabola  a  line  is  drawn  perpendicular  to 
any  tangent.  Required  the  locus  of  the  intersection  of  this  line  and  the  ordinate 
through  the  point  of  contact  of  the  tangent. 

Let  Pi(xi,  2/i)  be  any  point  of  the  parabola  y^  =  ipx  (fig.  187),  PiT  the 
tangent  at  Pi,  and  OT  the  perpendicular  to  PiT  from  the  vertex  0.  Then  the 
equation  of  PiT  is 

yiy  =  2p{x  +  xi),  (1) 

2/1 


and  the  equation  of  0  T  is 


y  =  -  —X. 
2p 


The  equation  of  the  ordinate  MiPi  through  Pi  is 

X  =  Xi. 


(2) 


(8) 


320    PARAMETRIC  REPRESENTATION  OF  CURVES 

If  P(x,  y)  is  the  point  of  intersection  of  (2)  and  (3),  P  is  any  point  of  the 
locus,  and  (2)  and  (3)  form  the  implicit  parametric  representation  of  the  locus 
in  terms  of  the  parameters  Xi  and  yi.  Since  Pi  (Xi,  yi)  is  by  hypothesis  any 
point  of  the  parabola,  its  coordinates  satisfy  the  equation  of  the  parabola,  and 
the  parameters  Xi  and  yi  satisfy  the  equation 

y^  =  4pxi.  (4) 


Fig.  187 

Solving  (2)  and  (3)  for  Xi  and  yi  and  substituting  their  values  in  (4),  we 
thereby  eliminate  them  and  have,  as  the  Cartesian  equation  of  the  locus. 


2/2  =  _  x8. 
P 


(5) 


From  the  form  of  the  equation  tlie  locus  is  seen  to  be  a  semicubical  parabola. 
It  may  be  added  that  the  explicit  parametric  representation  of  the  locus  is 

-  Vi^i 
2p 


readily  found  to  be  x  =  Xj  and  y  = 


,  where  y^  -  4pxi. 


LOCUS  PROBLEMS 


321 


Ex.  2.    Locus  of  the  middle  points  of  chords  of  an  ellipse,  drawn  through  one 
end  of  its  major  axis. 

Let  the  ellipse  be 1 =  1  (fig.  188),  and  Pi(a;i,  yi)  be  any  point  of  the 

a^      b^ 
ellipse.    Then  APi  is  any  chord  through  A,  and  P(x,  y),  its  middle  point,  is 
any  point  of  the  required  locus.    Since  the  coordinates  of  A  are  (a,  0),  by  §  18 


xi  +  a 


and 


y  = 


y\ 


Then  (1)  and  (2)  are  the  explicit  para- 
metric representations  of  the  locus  in 
terms  of  the  parametere  Xi  and  y^ 
which  satisfy  the  equation 

(3) 


— +  — =  1, 

02  62 


Fig,  188 


since  Pi  is  any  point  of  the  ellipse. 

To  find  the  Cartesian  equation  of  the  locus,  we  substitute  in  (3)  the  values  of 
Xi  and  yi  from  (1)  and  (2).    The  result  is 

Accordingly  the  locus  is  an  ellipse  with  its  center  at  ( -,  0  )  and  its  semiaxes 
equal  respectively  to  -  and  -  ■ 

Ex.  3.    Locus  of  the  point  of  intersection  of  tangents  at  the  ends  of  conjugate 
diameters  of  an  ellipse. 


"~^ 

^/-"""T^:^-- 

^ 

\(i) 

y^    ^icC"^ 

^ 

^ 

>^^r^\ 

0- 

^ 

J      J 

\s,^^   Jir-——— __ 

-^ 

-^^2  ^y 

^--^ 

X 


Fig. 189 


X2        V^ 

Let  the  ellipse  be 1-^  =  1  (fig.  189),  and  OAi  and  OBi  be  any  two  con- 

a2      62  /  6    \ 

jugate  diameters.    If  Ax  is  (xi,  yj),  -Bi  is  ( ^,  — ^-\  by  ICx.  2,  §  146. 

\       6        a  J 


322    PARAMETRIC  REPRESENTATION  OF  CURVES 


Then  the  tangents  at  Ai  and  Bx  will  be  respectively 


and 
where 


Solving  (1)  and  (2),  we  find      z  = 


y  = 


62 

ab     ^  ab~    ' 

a2      62 
6xi  —  ayi 


bxi  +  ayi 


(1) 
(2) 

(3) 
(4) 
(6) 


as  the  explicit  parametric  representations  of  the  locus. 

If  we  write  (4)  and  (5)  in  forms  bz  —  bxi  —  ayi  and  ay  =  bxi  +  ayi  respec- 
tively, and  square  and  add,  we  have 

62a;2  +  a22/2  =  2  (62x2  +  a^^), 
or  62x2  +  a2y2  =  2  a262,  (6) 

by  virtue  of  (3). 

y^ 


As  (6)  may  be  written 


+ 


=  1,  we  see  that  the  required  locus  is 


(a  V2)2      (6  V2)2 
an  ellipse,  concentric  with  the  given  ellipse  and  with  the  semiaxes  a  V2  and  6  V2. 

Ex.  4.  P1P2  is  any  chord  of  an  ellipse  perpendicular  to  its  major  axis  A\Ai. 
Find  the  locus  of  point  of  intersection  of  AiPi  and  A2P2. 


Fig.  190 

x2      y^ 
Let  the  ellipse  be 1-  —  =  1  (fig-  190),  and  the  coordinates  of  Pi  and  P2  be 

a2      62 
respectively  (xi,  yi)  and  (xi,  —  yi).    Then  the  equation  of  AiPi  and  A2P2  are 
respectively  „ 


y 


xi  +  a 

Vi 
a  —  xi 


{x  +  a), 
(z  -  a). 


(1) 

(2) 


PROBLEMS  323 

which  are  accordingly  the  implicit  parametric  representation  of  the  locus.    The 
parameters  Xi  and  yx  satisfy  the  equation 

^  +  g  =  l.  (8) 


Taking  the  product  of  (1)  and  (2),  we  have 

y^  =  -lL^{x^-a\  (4) 

which  may  be  written  y"^  —  —  (x2  —  dF),  (6) 

a2 


by  virtue  of  (3). 
As  (6)  may 
hyperbola  concentric  with  the  ellipse  and  having  the  same  semiaxes. 


As  (6)  may  be  written =  1,  we  see  that  the  required  locus  is  an 

a2      62 


PROBLEMS 

1.  Show  that  X  =  pt^,  y  =  2pt  are  parametric  equations  of  the  parabola. 

2.  Find  the  equations  of  the  tangent  and  the  normal  to  the  parabola  when 
the  equations  of  the  parabola  are  as  in  problem  1. 

3.  Find  the  parametric  equations  of  the  parabola  when  the  parameter  is 
the  slope  of  a  line  through  the  vertex. 

4.  Find  the  equations  of  the  tangent  and  the  normal  to  a  parabola  when 
the  equations  of  the  curve  are  as  in  problem  3. 

5.  Find  the  parametric  equations  of  the  ellipse  when  the  parameter  is  the 

slope  of  a  straight  line  through  the  center. 

6.  Find  the  parametric  equations  of  the  ellipse  when  the  parameter  is  the 
slope  of  a  straight  line  through  the  left-hand  vertex. 

7.  Find  the  parametric  equations  of  the  cissoid  when  the  parameter  is  the 
angle  AOP  (fig.  91). 

8.  Show  that  x  =  t,  y  — are  parametric  equations  of  the  witch. 

a2  +  <2 

2  (I  2  (I 

9.  Show  that  x  = ,  y  — —  are   parametric   equations  of  the 

\^t^  <(l  +  «2)  ^  ^ 

cissoid.    What  is  the  geometric  significance  of  i  ? 

10.  Find  the  equation  of  the  tangent  to  the  cissoid  if  the  equations  of  the 
curve  are  as  in  problem  9. 


324    PARAMETRIC  REPRESENTATION^  OF  CURVES 

Find  the  Cartesian  equations  of  each  of  the  following  curves : 

. ,  o   <2  +  2  a   t^  +  2t 

11.  X  =  -  ■ ,  y  =  ~  ■ — . 

2    P+\  2     i^  +  1 

12.  X  — ,  y  = 

\-^t^  1  +  «3 

,o                        *:^  ,  kH 

l«j.  x  =  a-\ -^,  y  =  at  + 


a(l  +  <2)  a(l  +  «2) 


,.  l  +  2«  l  +  < 

14.  x  = ,  y  = 

15.  X  = -,  y 


t-1  t^-1 

.„  e' +  e-'  e' —  C-' 

16,  X  = ,  y  = . 

{e'-2e-'y^  {e'-2e-'f 

17.  x  =  t^  +  St  +  2^  y  =  t^-l. 

,  o                      ct  ct^ 

^°-  ^  =  '. \ ; ;r'  y 


(a  +  U)  (1  +  «2)     "       (a  +  6t)  (1  +  «2) 

19.  Eliminate  <  from 

X  _  cos  <  —  sin  <  y  _  cos  <  +  sin  ^ 

a  e'  a  e' 

and  prove  that  the  curve  represented  is  a  logarithmic  .spiral  (§  178). 

20.  Let  0  be  the  center  of  a  circle  with  radius  a,  ^  a  fixed  point,  and  B  a 
moving  point  on  the  circle.  If  the  tangent  at  B  meets  the  tangent  at  A  in  C, 
and  P  is  the  middle  point  of  BC,  find  the  equations  of  the  locus  of  P  in  para- 
metric form,  using  the  angle  AOB  as  the  arbitrary  parameter,  OA  as  the  axis 
of  a;,  and  0  as  the  origin.    Also  find  the  Cartesian  equation  of  the  locus. 

21.  OBCD  is  a  rectangle  with  OB  =  a  and  BC  =  c.  Any  line  is  drawn 
through  C,  meeting  OB  in  E,  and  the  triangle  EPO  is  constructed  so  that  the 
angles  CEP  and  EPO  are  right  angles.  Find  the  parametric  equations  of  the 
locus  of  P,  using  the  angle  DOP  as  the  parameter,  OB  as  the  axis  of  x,  and  0 
as  the  origin.    Find  also  the  Cartesian  equation  of  the  locus. 

22.  Let  AB  be  a  given  line,  0  a  given  point,  a  units  from  AB,  and  k  a 
given  constant.  On  any  line  through  0,  meeting  AB  in  M,  take  P  so  that 
OM  ■  MP  =  A:2.  Find  the  parametric  equations  of  the  locus  of  P,  using  0  as  the 
origin,  the  perpendicular  from  O  to  ^Z?  as  the  axis  of  x,  and  the  angle  between 
OX  and  OP  as  the  parameter.    Also  find  the  Cartesian  equation. 

23.  A  and  B  are  two  points  on  the  axis  of  ?/  at  a  distance  —  a  and  4-  a 
respectively  from  the  origin.  AH  ifi  any  line  through  A  meeting  the  axis  of  x 
at  H.  BK  is  the  perpendicular  from  B  on  AH,  meeting  it  at  K.  Tiirough  K  a 
line  is  drawn  parallel  to  the  axis  of  x  and  through  H  a  line  is  drawn  parallel  to 
the  axis  of  y.  These  lines  meet  in  P.  P"ind  the  parametric  equations  of  the  locua 
of  P,  using  the  angle  BAK  as  the  parameter.    Also  find  the  Cartesian  equation. 


PROBLEMS  325 

24.  Let  OA  be  the  diameter  of  a  fixed  circle  and  LK  tlie  tangent  at  A. 
From  0  draw  any  line  intersecting  the  circle  at  B  and  LK  at  C,  and  let  P  be 
the  middle  point  of  BC.  Find  the  parametric  equations  of  the  locus  of  P, 
using  the  angle  A  OP  as  the  parameter,  OA  as  the  axis  of  y,  and  O  as  the  origin. 
Find  also  the  Cartesian  equation. 

25.  Show  that  the  tangent  to  the  ellipse  at  any  point  and  the  tangent  to  the 
auxiliary  cii'cle  at  the  corresponding  point  pass  through  the  same  point  of  the 
major  axis. 

26.  Prove  that  the  eccentric  angles  of  the  ends  of  a  pair  of  conjugate 
diameters  of  an  ellipse  differ  by  —  • 

27.  Show  that  the  perpendicular  from  either  focus  upon  the  tangent  at  any 
point  of  the  auxiliary  circle  of  an  ellipse  equals  the  focal  distance  of  the  corre- 
sponding point  of  the  ellipse. 

28.  Q  is  the  point  on  the  auxiliary  circle  of  the  ellipse,  corresponding  to 
the  point  P  of  the  ellipse.  The  straight  line  through  P  parallel  to  OQ  meets 
OX  at  L  and  OY  at  M.    Prove  PL  =  b,  and  PM  =  a. 

29.  Find  the  equation  of  the  tangent  at  any  point  of  an  ellipse  in  terms  of 
the  eccentric  angle  at  that  point. 

30.  What  elevation  must  be  given  to  a  gun  to  obtain  the  maximum  range 
on  a  horizontal  line  passing  through  the  muzzle  of  the  gun  ?  (In  this  and  the 
following  examples  the  resistance  of  the  air  and  the  effect  of  all  forces  except 
gravity  are  neglected.) 

31.  What  elevation  must  be  given  to  a  gun  to  obtain  a  maximum  range  on 
an  oblique  line  passing  through  the  muzzle  of  the  gun  and  making  an  angle  ^ 
with  the  horizontal  ? 

32.  What  elevation  must  be  given  to  a  gun  that  the  projectile  should  pass 
through  a  point  in  the  horizontal  line  passing  through  the  muzzle  and  6  units 
from  it  ? 

33.  A  gun  stands  on  a  cliff  h  units  above  the  water.  What  elevation  must 
be  given  to  the  gun  that  the  projectile  may  strike  a  point  Tn  the  water  b  units 
from  the  base  of  the  cliff  ? 

34.  Find  the  parametric  equations  of  the  curve  described  by  any  point  in 
the  connecting  rod  of  a  steam  engine. 

35.  If  a  circle  rolls  on  the  inside  of  a  fixed  circle  of  twice  its  radius,  what  is 
the  form  of  the  curve  generated  by  a  point  of  the  circumference  of  the  rolling 
circle  ? 

36.  Show  that  the  hypocycloid  generated  when  the  rolling  circle  has  J  the 
radius  of  the  fixed  circle  has  the  Cartesian  equation  x*  -f-  y^  =  6'. 

37.  If  a  wheel  rolls  with  constant  angular  velocity  on  a  straight  line, 
required  the  velocity  of  any  point  on  its  circumference  ;  also  of  any  point  on 
one  of  the  spokes. 


326     PARAMETRIC  REPRESENTATION  OF  CURVES 

38.  If  a  wheel  rolls  with  constant  angular  velocity  on  the  circumference 
of  a  fixed  wheel,  find  the  velocity  of  any  point  on  its  circumference  and  on  its 
spoke. 

39.  Show  that  the  highest  point  of  a  wheel  rolling  with  constant  velocity 
on  a  road  moves  twice  as  fast  as  each  of  the  two  points  in  the  rim  whose  dis- 
tance from  the  ground  is  lialf  the  radius  of  the  wheel. 

40.  If  a  string  is  unwound  from  a  circle  with  constant  velocity,  find  the 
velocity  of  the  end  in  the  path  described. 

41.  AB  and  CD  are  perpendicular  diameters  of  a  circle  of  radius  E. 
AM  is  a  chord  of  the  circle,  rotating  about  A  so  that  the  angle  BAM  varies 
uniformly.  AM  is  extended  to  N  so  that  J!f^=the  chord  MB.  Find  the 
path  of  N,  the  velocity  of  N  in  its  path,  and  the  components  of  the  velocity 
respectively  parallel  to  AB  and  CD. 

42.  0,  CK,  0"  are  three  points  on  a  straight  line  and  0"(y  =  ^0(7.  LK is 
drawn  through  C  pei-pendicular  to  00" ^  and  any  point  M  is  taken  on  LK. 
From  M  a  straight  line  is  drawn  perpendicular  to  0"Jf,  and  through  0  a 
straight  line  is  drawn  parallel  to  0"M.  These  lines  intersect  in  P.  Required 
the  locus  of  P. 

43.  0  is  a  fixed  point  and  LK  a  fixed  straight  line.  Any  point  M  is  taken 
on  LK,  and  the  line  OM  is  drawn  and  prolonged  to  P  so  that  OM  ■  OP  =  k^, 
where  A;  is  a  constant.    Find  the  locus  of  P. 

44.  Show  that  the  locus  of  points  symmetrical  to  the  vertex  of  a  parabola 
with  respect  to  its  tangent  lines  is  a  cissoid. 

45.  Let  OA  be  the  diameter  of  any  circle  and  LK  the  tangent  at  A, 
Through  0  draw  any  line  intersecting  the  circle  in  D  and  LK  in  E.  Lay  off  on 
OE  produced  the  distance  EP  =  OD,  and  find  the  locus  of  P. 

46.  Let  a  circle  with  center  at  0  intersect  the  axis  of  y  at  J.  and  the  axis 
of  X  at  C.  Take  two  points  G  and  E  on  the  circle  equidistant  from  A.  If  the 
ordinate  of  G  intersects  the  line  CE  in  P,  prove  that  the  locus  of  P  is  a  cissoid. 

47.  From  a  point  a  units  from  the  axis  of  x  lines  are  drawn  to  OX,  and 
from  the  point  where  each  line  meets  the  axis  a  line  of  tlie  same  length  is 
drawn  at  right  angles  to  the  first  line.  Find  the  equation  of  the  locus  of  the 
end  of  this  last  line. 

48.  OA  is  a  diameter  of  a  circle  and  LK  the  tangent  at  A.  Through  0  any 
line  is  drawn  meeting  the  circle  in  B  and  LK  in  C.  Through  B  a  line  is  drawn 
perpendicular  to  OA  and  meeting  it  in  M.  Finally  MB  is  prolonged  to  P  so 
that  MP  =  AC.    Find  the  locus  of  P. 

49.  Find  the  path  described  by  any  point  of  a  tangent  line  which  rolls  upon 
a  circle  without  slipping. 

50.  CD  is  perpendicular  to  OX  and  distant  a  units  from  O.  Through  A ,  any 
point  on  CD,  a  straight  line  OA  is  drawn,  and  from  A  a  perpendicular  is  drawn 


PEOBLEMS  327 

to  OA,  intersecting  OX  at  B.  From  B  a  straight  line  is  drawn  parallel  to  OY, 
intersecting  OA  at  P.  If  m  denotes  the  slope  of  OA,  find  the  parametric  and 
the  Cartesian  equations  of  the  locus  of  P, 

51.  Prove  that  the  pedal  of  a  parabola  with  respect  to  any  point  is  a  cubic 
curve  which  passes  through  that  point. 

52.  Prove  that  the  pedal  of  the  ellipse \-  —  =  \  with  respect  to  the  center 

is  the  curve  (x2  +  y2)2  =  a'^n^p.  +  52^2.  <^^      ^ 

53.  A  line  of  constant  length  k  moves  with  its  extremities  on  the  two  axes 
of  coordinates.    Find  the  locus  described  by  any  point  of  the  line. 

54.  A  straight  line  has  its  extremities  on  the  coordinate  axes  and  passes 
through  a  fixed  point.    Find  the  locus  of  its  middle  point. 

55.  If  the  ordinate  NP  of  an  hyperbola  be  produced  to  Q,  so  that  NQ,  =  FP, 
find  the  locus  of  Q. 

56.  Find  the  locus  of  the  points  of  intersection  of  normals  at  corresponding 
points  of  the  ellipse  and  the  auxiliary  circle. 

57.  P  is  any  point  of  a  parabola,  A  the  vertex,  and  through  A  a  straight 
line  is  drawn  perpendicular  to  the  tangent  at  P.  Find  the  locus  of  the  point  of 
intersection  of  this  line  with  the  diameter  through  P,  and  also  the  locus  of  the 
point  of  intersection  of  this  line  with  the  ordinate  through  P. 

58.  Two  equal  parabolas  have  their  axes  parallel  and  a  common  tangent  at 
their  vertices,  and  straight  lines  are  drawn  parallel  to  the  axes.  Show  that  the 
locus  of  the  middle  points  of  the  parts  of  the  lines  intercepted  between  the 
curves  is  an  equal  parabola. 

59.  Find  the  locus  of  the  intersection  of  the  ordinate,  produced  if  necessary, 
of  any  point  on  an  ellipse  with  the  perpendicular  from  the  center  upon  the 
tangent  at  that  point. 

60.  Two  parabolas  have  the  same  axis,  and  tangents  are  drawn  from  points 
on  the  first  to  the  second.  Prove  that  the  middle  points  of  the  chords  of  con- 
tact with  the  second  lie  on  a  parabola. 

61.  Chords  of  an  ellipse  are  passed  through  a  fixed  point.  Find  the  locus  of 
their  middle  points. 

62.  From  a  point  P  on  an  ellipse  straight  lines  are  drawn  to  the  vertices  A 
and  A\  and  from  A  and  A'  straight  lines  are  drawn  perpendicular  to  AP  and 
A'P.    Show  that  the  locus  of  their  point  of  intersection  is  an  ellipse. 

63.  Show  that  the  locus  of  the  point  of  intersection  of  two  tangents  to  a 
parabola,  the  ordinates  of  the  points  of  contact  of  which  are  in  a  constant 
ratio,  is  a  parabola. 

64.  If  the  tangent  to  the  parabola  y^  =  4px  meets  the  axis  at  T  and  the 
tangent  at  the  vertex  A  at  J5,  and  the  rectangle  TABQ  is  completed,  show 
that  the  locus  of  Q  is  the  parabola  y^  +  px  =  0. 


328    PARAMETRIC  REPRESENTATION  OF  CURVES 

65.  Find  the  locus  of  the  feet  of  the  perpendiculars  from  the  focus  to  the 
normals  of  the  parabola  y^  =  4px. 

66.  Show  that  perpendicular  normals  to  the  pai'abola  y-  =  4px  intersect  on 
the  curve  y'^  =  px  —  Sp^. 

67.  Find  the  locus  of  the  intersection  of  a  pair  of  perpendicular  tangents  to 
an  hyperbola. 

68.  Two  tangents  to  an  ellipse  are  so  drawn  that  the  product  of  their  slopes 
is  constant.  Show  that  the  locus  of  their  point  of  intersection  is  an  ellipse  or 
an  hyperbola  according  as  the  product  is  negative  or  positive. 

69.  Prove  that  the  locus  of  the  point  of  intersection  of  two  tangents  to  a 
parabola  is  a  straight  line  if  the  product  of  their  slopes  is  constant. 

70.  Find  the  locus  of  the  foot  of  the  perpendicular  from  either  focus  of  an 
hyperbola  to  any  tangent. 

71.  Let  AB  be  the  diameter  of  a  circle  and  O  its  center.  Let  NQ  be  the 
ordinate  of  a  point  Q  on  the  circle  and  P  another  point  of  the  circle,  so  related 
to  Q  tjiat  OP  revolves  uniformly  from  OA  through  a  right  angle  in  the  same 
time  that  QN  travels  at  a  unifoi-m  rate  from  A  to  0.  If  OP  and  QN  intersect 
in  R,  find  the  locus  of  R. 

72.  Find  the  equations  of  the  cycloid  when  the  tangent  at  its  highest  point 
is  the  axis  of  x,  the  normal  at  the  vertex  is  the  axis  of  y,  and  the  angle  0  is 
the  angle  through  which  the  radius  has  rotated  after  passing  through  the 
highest  point. 

73.  Prove  that  the  area  of  an  arch  of  the  cycloid  above  the  axis  of  x  is 
three  times  the  area  of  the  rolling  circle. 

74.  Prove  that  for  a  cycloid  —  =  2  a  sin  - ,  and  thence  find  its  length  from 
.  dd>  2 

cusp  to  cusp.  ^ 

75.  Show  that  for  an  epicycloid  —  =  2(a  +  6)sin  — ,  and  thence  find  its 
length  from  cusp  to  cusp,  ^ 


CHAPTEE  XV 
POLAR  COORDINATES 

177.  Coordinate  system.  So  far  we  have  determined  the  posi- 
tion of  a  point  in  the  plane  by  two  distances,  x  and  y.  We  may, 
however,  use  a  distance  and  direction,  as  follows : 

Let  0  (fig.  191),  called  the  origin  or  foh,  be  a  fixed  point,  and 
OM,  called  the  initial  line,  be  a  fixed  line.  Take  P  any  point  in 
the  plane  and  draw  OP.  Denote  OP  by  r  and  the  angle  MOP  by  0. 
Then  r  and  Q  are  called  the  'polar  coordinates  of  the  point  P(r,  6), 
and  when  given  will  completely  determine  P. 


Fig.  191 

For  example,  the  point  (2,  15°)  is  plotted  by  laying  off  the 
angle  MOP  =  15°  and  measuring  OP  =  2. 

OP,  or  r,  is  called  the  raditis  vector  and  0  the  vectorial  angle  of  P. 
These  quantities  may  be  either  positive  or  negative.  A  negative 
value  of  0  is  laid  off  in  the  direction  of  the  motion  of  the  hands 
of  a  clock,  a  positive  angle  in  the  opposite  direction.  After  the 
angle  0  has  been  constructed,  positive  values  of  r  are  measured 
from  0  along  the  terminal  line  of  0,  and  negative  values  of  r  from 
O  along  the  backward  extension  of  the  terminal  line.  It  follows 
that  the  same  point  may  have  more  than  one  pair  of  coordinates. 

320 


330 


POLAR  COORDINATES 


Thus  (2,  195°),  (2,  -  165°),  (-  2,  15°),  and  (-  2,  -  345°)  refer  to 
the  same  point.  In  practice  it  is  usually  convenient  to  restrict  6 
to  positive  values. 

Plotting  in  polar  coordinates  is  facilitated  by  using  paper  ruled 
as  in  figs.  192  and  193.  The  angle  6  is  determined  from  the  num- 
bers at  the  ends  of  the  straight  lines,  and  the  value  of  r  is  counted 
off  on  the  concentric  circles,  either  towards  or  away  from  the  num- 
ber which  indicates  6,  according  as  r  is  positive  or  negative. 

When  an  equation  is  given  in  polar  coordinates  the  correspond- 
ing curve  may  be  plotted  by  giving  to  6  convenient  values,  com- 
puting the  corresponding  values  of  r,  plotting  the  resulting  points, 
and  drawing  a  curve  through  them. 

Ex.  1.    r  =  a  costf. 

a  is  a  constant  which  may  be  given  any  convenient  value.  We  may  then  find 
from  a  table  of  natural  cosines  the  value  of  r  which  corresponds  to  any  value  of  0. 


165) 


180 


195 


By  plotting  the  points  corresponding  to  values  of  6  from  0°  to  90°  we  obtain  the 
9,TcABC0  (fig.  192).    Values  of  6  from  90°  to  180°  give  the  arc  ODEA.    Values  of 


GRAPHS 


331 


e  from  180°  to  270°  give  again  the  arc  ABCO,  and  those  from  270°  to  360°  give  the 
arc  ODEA.  Values  of  d  greater  than  300°  can  clearly  give  no  points  not  already- 
found.    The  curve  is  a  circle  (§  184). 

Ex.  2.    r  =  o  sin  3  6. 

As  0  increases  from  0°  to  30°,  r  increases  from  0  to  a ;  as  ^  increases  from  30° 
to  60°,  r  decreases  from  a  to  0;  the  point  P{r,  6)  traces  out  the  loop  OAO  (fig. 
193).    As  e  increases  from  60°  to  90°,  r  is  negative  and  decreases  from  0  to  —  a  •, 


5  if 


as  6  increases  from  90°  to  120°,  r  increases  from  —  a  to  0 ;  the  point  (r,  0)  traces 
out  the  loop  OBO.  As  6  increases  from  120°  to  180°,  the  point  (r,  0)  traces  out  the 
loop  OCO.  Larger  values  of  0  give  points  already  found,  since  sin  3  (180°  +  0) 
=  —  sin  3  $.  The  three  loops  are  congruent  because  sin  3  (60°  +  0)  =  —  sin  3  0. 
This  curve  is  called  a  rose  of  three  leaves. 


178.  The  spirals.  Polar  coordinates  are  particularly  well 
adapted  to  represent  certain  curves  called  spirals,  of  which  the 
more  important  follow. 


332  POLAR  COORDINATES 

Ex.  1.    The  spiral  of  Archimedes, 

r  =  ae. 

In  plotting  6  is  usually  considered  in  circular  measure.    When  $  =z  0,  r  =  0, 
and  as  6  increases  r  increases,  so  that  the  curve  winds  infinitely  often  around 


Fig.  Voi 

the  origin  while  receding  from  it  (fig.  194).  In  the  figure  the  heavy  line  repre- 
sents the  portion  of  the  spiral  corresponding  to  positive  values  of  0,  and  the 
dotted  line  the  portion  corresponding  to  negative  values  of  6. 

Ex.  2.     The  hyperbolic  spiral, 

rd  =  a, 
a 


L- 


K 


Fig.  19.J 

As  0  increases  indefinitely  r  approaches  zero.    Hence  the  spiral  winds  infi- 
nitely often  around  the  origin,  continually  approaching  it  but  never  reaching  it 


THE  SPIRALS  333 

(fig.  195).    As  0  approaches  zero  r  increases  without  limit.    If  P  is  a  point  on 
the  spiral  and  NP  the  perpendicular  to  the  initial  line, 

■»TT,          •    «         sin^ 
NP  =  rsmd  =  a 

e 

Hence  as  6  approaches  zero  as  a  limit,  NP  approaches  a  (§  161).  Therefore 
the  curve  comes  constantly  nearer  to,  but  never  reaches,  the  line  LK,  parallel 
to  OM  at  a  distance  a  units  from  it.  This  line  is  therefore  an  asymptote.  In 
the  figure  the  dotted  portion  of  the  curve  corresponds  to  negative  values  of  0. 

Ex.  3.    The  logarithmic  spiral, 

r  =  e"*. 

When  6  =  0,  r  =  1.  As  ^  increases  r  increases,  and  the  ciirve  winds  around 
the  origin  at  increasing  distances  from  it  (fig.  196).  When  0  is  negative  and 
increasing  numerically  without  limit,  r  approaches  zero.  Hence  the  curve 
winds  infinitely  often  around  the  origin,  continually  approaching  it.  The 
dotted  line  in  the  figure  corresponds  to  negative  values  of  0. 


Fig.  196 

A  property  of  this  spiral  is  that  it  cuts  the  radii  vectors  at  a  constant  angle. 
The  student  may  prove  this  after  reading  §  187. 

We  shall  now  give  examples  of  the  derivation  of  the  polar  equa- 
tion of  a  curve  from  the  definition  of  the  curve. 


334 


POLAR  COORDINATES 


179.  The  conchoid.  Take  a  fixed  point  0  (fig.  197)  and  a  fixed 
straight  line  BC.  Through  0  draw  any  line  OR  intersecting  BC 
in  D,  and  on  OB  lay  off  a  constant  distance  DP  or  DQ,  measured 
from  D  in  either  direction.  The  locus  of  P  and  ^  is  a  curve  called 
the  conchoid. 

From  the  definition  the  conchoid  consists  of  two  parts,  one 
generated  by  P,  the  other  by  Q.    We  may  obtain  the  whole  curve, 


Fig.  198 


however,  by  allowing  the  line  OR  to  revolve  in  the  positive  direc- 
tion through  an  angle  of  360°  and  always  laying  off  the  distance  h, 
measured  from  D  in  the  direction  of  the  terminal  line  of  the  angle 
AOR.  Then  if  AOR  is  in  the  first  quadrant,  we  obtaia  the  upper 
half  of  the  curve  described  by  P ;  if  ^  OR  is  in  the  second  quad- 
rant, we  have  the  lower  half  of  the  curve  described  by  Q;  HA  OR 
is  in  the  third  quadrant,  we  have  the  upper  half  of  the  curve 


THE  CONCHOID 


335 


described  by  Q ;  and  if  A  OB  is  in  the  fourth  quadrant,  we  have 
the  lower  half  of  the  curve  described  by  P. 

To  find  its  polar  equation,  take  0  as  the  origin  and  the  line  OA 
perpendicular  to  BC  as  the  initial  line.  Let  OA  =  a  and  the  con- 
stant distance  DP  =  h. 

Call  the  coordinates  of  P  (r,  6),  where  6  =A  OR.  When  ^  is  in 
the  first  or  the  fourth  quadrant,  r  =  OD  +  DP  =  OD  +  h ;  when  6 
is  in  the  second  or  the  third  quadrant,  r  =  —  OD  -\-DQ  =  —  OD  +  h. 


Fig.  199 


But  OD  =  a  sec  d  when  ^  is  in  the  first  or  the  fourth  quadrant ; 
and  OD  =  —  a  sec  6  when  ^  is  in  the  second  or  the  third  quadrant. 
Hence  for  all  points  on  the  conchoid 

r  =  a  sec  ^  +  &. 

The  conchoid  has  three  shapes  according  as  a  >  i  (fig.  197), 
a  =  6  (fig.  198),  a<l  (fig.  199).  If  &  =  0,  the  conchoid  becomes 
the  straight  line  BC  and  its  equation  becomes  r  =  a  sec  ^,  the 
equation  of  the  straight  line  (§  183). 


336 


POLAR  COORDINATES 


180.  The  limacon.  Through  any  fixed  point  0  (fig.  200)  on  the 
circumference  of  a  fixed  circle  draw  any  line  cutting  the  circle 
again  at  Z>,  and  lay  off  on  this  line  a  constant  length  measured 
from  D  in  either  direction.  The  locus  of  the  points  P  and  ^  thus 
found  is  a  curve  called  the  limagon. 

Take  0  as  the  pole,  the  diameter  OA  as  the  initial  line  of  a  sys- 
tem of  polar  coordinates,  and  call  the  diameter  of  the  circle  a  and 


Fig.  200 


the  constant  length  h.  Then  it  is  clear  that  the  entire  locus  can 
be  found  by  causing  OD  to  revolve  through  an  angle  of  360°  and 
laying  off  DP  =  h  always  in  the  direction  of  the  terminal  line 
of  ADD. 

Let  the  coordinates  of  P  be  (r,  6),  where  0=AOD.  Then 
r  =  OD  +  DP  when  6  is  in  the  first  or  the  fourth  quadrant, 
and  r  =  —  OD  +  DP  when  6  is  in  tlie  second  or  the  third 
quadrant.  But  it  appears  from  the  figure  that  OD  =  OA  cos  6 
when  6  is  in  the  first  or  the  fourth  quadrant,  and  OD  =  —  OA  cos  d 


THE  LIMAgON 


337 


when  6  is  in  the  second  or  the  third  quadrant.    Hence  for  any 
point  on  the  limagon 

r  =  a  cos  6  +  1). 

In  studying  the  shape  of  the  curve  there  are  three  cases  to  be 
distinguished. 


O^cos\-i) 


Fig.  201 


1.  h>a.   r  is  always  positive  and  the  curve  appears  as  in  fig.  200. 

2.  b  <  a.    r    is    positive    when    cos  d  > '   negative   when 

7  h  ^ 

cos  6  < '  and  zero  when  cos  0  = •    The  curve  appears  as 

a  a  ^'^ 

in  fig.  201. 

3.  h  =  a.    The  equation  now  becomes 

r  =  a  (cos  ^  +  1)  =  2  a  cos^  -  ■ 

r  is  positive  except  when  6  =  180°,  when  it  is  zero.    The  curve 
appears  as  in  fig.  202  and  is  called  the  cardioid. 


338 


POLAE  COORDmATES 


The  cardioid  is  an  epicycloid  for  wMch  the  radii  of  the  fixed  and 
the  rolling  circles  are  the  same.  The  proof  of  this  is  left  to  the 
student. 


Fig.  202 

181.  The  ovals  of  Cassini.    If  a  point  moves  so  that  the  product 
of  its  distances  from  two  fixed  points  is  constant,  it  generates  a 

r 


Fig.  203 


curve  called  an  oval  of  Cassini.    Let  F^  and  F^  (fig.  203)  be  the 
two  fixed  points,  called  the  foci,  and  If  the  constant  product  of 


THE  OVALS  OF  CASSINI  339 

the  distances  of  a  point  of  the  curve  from  F^  and  F^.  Take  F^F^ 
as  the  initial  line  and  the  point  .0,  halfway  between  F^  and  F^,  as 
the  pole  of  a  system  of  polar  coordinates,  and  let  P  be  a  point  on 
the  curve.    Then,  by  definition, 

F^P  .  F^P  =  l\  (1) 

By  trigonometry, 

:^'  =  OB'^j^OF^-  2  OP  •  OF^  cosF^OP  =  r'+  €0"+  2  ra  cos(9, 

where  (r,  6)  are  the  coordinates  of  P  and  2a=  F^F^.    Also 

TJ^''  =  op'^j^OF^-  2  OP  •  OF^  QosF^OP  =  r'+  a"-  2  ra  cos^. 

Substituting  in  (1),  we  have 

(r2+ay_4aV'cos'(9  =  J*, 
which  is  the  same  as 

/_  2  aV  cos  2  (9  +  a*-  5*  =  0.  (2) 

To  determine  the  form  of  the  curve,  it  is  convenient  to  solve  (2) 
for  r^,  obtaining 

r"  =  a^cos2e±  ^a*  cos'2  0 -{a*-b*).  (3) 

We  have,  then,  three  cases  to  consider 

1.  a^  <  If.  The  quantity  under  the  radical  sign  in  (3)  is  posi- 
tive and  greater  than  a*  cos^  2  6  for  all  values  of  6.  Therefore  r^ 
in  (3)  has  two  real  values,  one  positive  and  one  negative.  Conse- 
quently r  has  two,  and  only  two,  real  values  equal  in  magnitude 
and  opposite  in  sign.  The  curve  therefore  consists  of  a  single  oval, 
symmetric  with  respect  to  the  origin  (fig.  203). 

a*—  h* 

2.  a^  >  If.  When  cos*^  2  0  > —  the  quantity  under  the  rad- 
ical sign  in  (3)  is  positive  and  less  than  a*  cos^  2  6.  Hence  for 
these  values  of  6  there  are  two  real  positive  values  of  r^  and  there- 
fore four  real  values  of  r,  two  positive  and  two  negative.    When 

a^—h*  .  . 

cos^  2  0  < —  the  quantity  under  the  radical  sign  in  (3)  is 


340  POLAE  COOKDINATES 

negative,  and  hence  all  values  of  r  are  imaginary.   When  cos*^  2  Q 

a*  —  b*                                                                                   Va*  —  b* 
= —  there  are  two  real  values  of  ?',  namely  r  =  ± 


The  curve  consists  of  two  distinct  ovals  (fig.  204). 


-M 


Fig.  204 


3.  a^  =  b^.  Equation  (2)  then  factors  into  the  two  equations 
r^  =  0  and  r^—  2  a^  cos  2  ^  =  0.  But  r^  =  0  is  satisfied  only  by 
the  origin,  which  is  also  a  point  on  the  second  equation. 


Fig.  20.3 


Hence 


r'  =  2  a'  cos  2  d 


(4) 


is  the  full  equation  of  the  locus  in  this  case.    From  (4)  it  appears 
that  r  has  two  real  values  equal  in  magnitude  but  opposite  lq  sign 

when  0<^<->or— <6'<— ,or— <6'<2  7r.    Further, 
4  4  4  4 

r  =  0  when  6  =  —  ,  —^  >  —7-  ,  or  — — ;  and  r  is  imaginary  when 


4  4 

5  TT 


4  4  4  4 


4  4 


The  curve  appears  as  in  fig.  205 
and  is  given  the  special  name  of  the  lemniscate. 


CHANCxE  OF  cooedi:n:ates 


341 


182.  Relation  between  rectangular  and  polar  coordinates.    Let 

the  pole  O  and  the  initial  line  OM  of  a  system  of  polar  coordinates 
be  at  the  same  time  the  origin  and  the  axis  of  a;  of  a  system  of 
rectangular  coordinates.  Let  P  (fig.  206)  be  any  point  of  the  plane, 
{x,  y)  its  rectangular  coordinates,  and  {r,  6)  its  polar  coordinates. 
Then,  by  the  definition  of  the 
trigonometric  functions, 

X 

cos  6  =  -) 


sm  a  =  -  > 
r 


whence  follows,  on  the  one  hanC, 

X  =  r  cos  6, 
y  =  r  sin  Q, 

and,  on  the  other  hand. 


Fig.  206 


r  =  -Vaf  +  y ,         sin  ^  = 


y 


\^x^  +  y^ 


cos  tr  = 


Va;^ 


r 


(2) 


By  means  of  (1)  a  transformation  can  be  made  from  rectangular 
to  polar  coordinates,  and  by  means  of  (2)  from  polar  to  rectangular 
coordinates. 


Ex.  1.    The  equation  of  tlie  cissoid  (§  83)  is 

2/2  = 


2a  —  X 


Substituting  from  (1)  and  making  simple  reductions,  we  have  the  polar 

equation 

2asin2  6i 

r  = 

cos^ 


Ex.  2.    The  polar  equation  of  the  lemniscate  is 

r2  =  2  a2  cos  2  e. 

Placing  cos2^  =  cos2  0  —  sin^e  and  substituting  from  (2),  we  have  the  rec- 
tangular equation 

(x2  +  ?/2)2=:2a2(x2-2/2). 


342  POLAR  COORDINATES 

183.  The  straight  line.  Take  the  equation  of  the  straight  hne 
in  the  normal  form  x  cos a-{-y  sin a—p  =  0  and  substitute  the 
values  of  x  and  y  from  (1),  §  182.    There  results 

r  (cos  0  cos  a  +  sin  0  sin  a)  —  j?  =  0; 
whence  r  cos  {0  —  a)  =  p. 

A  reference  to  §  33  shows  that  {p,  a)  are  the  polar  coordinates 
of  the  point  in  which  the  normal  from  the  origin  meets  the  straight 
line.    If  a  =  0  and  p  =  a,  we  have  the  special  equation 

r  cos  0  =  a, 
or  r  =  a  sec  0, 

as  found  in  §  179. 

If  the  straight  liue  passes  through  the  origin,  ^  =  0.  The  equa- 
tion of  the  line  then  becomes 

cos  {6  —  a)=  0, 

or  simply  0  =  —  +  a, 

which  is  of  the  form  0  =  c. 

184.  The  circle.  If  (d,  e)  are  the  rectangular  coordinates  of  the 
center  of  the  circle  and  a  its  radius,  its  equation  is 

If  (h,  a)  are  the  polar  coordinates  of  the  center  and  (r,  0)  those 
of  any  point,  the  pole  and  the  initial  line  of  the  polar  coordinates 
being  the  origin  and  the  axis  of  x,  respectively,  of  the  rectangular 
system,  we  have,  by  (1),  §  182, 

x  =  r  cos  0,  y  =  rsm0, 

d  =  b  cos  a,  e  =  h  sin  a. 

We  obtain,  by  substitution, 

r'—2rb  (cos  0  cos  a  +  sin  ^  sin  a)+h^—  a^  =  0, 
or  r^-2rbcos(0-a)  +  b^-a^==O.  (1) 


P{r,e) 


THE  CONIC  343 

This  result  may  also  be  directly  obtained  from  fig.  207  by  noticing 
CF""  ='0C^ +0F'' -  2  OP  •  OC cos Foa 

"When  the  origin  is  at  the 
center  of  the  circle,  b  =  0, 
and  (1)  becomes  simply 

r  =  a.  (2) 

When  the  origin  is  on 
the  circle,  b  =  a,  and  (1) 
becomes 

r—2a  cos (^  —  a)  =  0,       ^^ 

which  may  be  written  ^'^-  -^^ 

r  =  aQ  cos  6  +  a^  sin  6,  (3) 

where  a^  and  a^  are  the  intercepts  on  the  lines  ^  =  0  and  6  =  — 

respectively. 

Wlien  the  origin  is  on  the  circle  and  the  initial  line  is  a 

diameter,  (3)  becomes 

r  =  «()  cos  6.  (4) 

When  the  origin  is  on  the  circle  and  the  initial  line  is  tangent 
to  the  circle,  (3)  becomes 

r  =  a^  sin  6.  (5) 

185.  The  conic,  the  focus  being  the  pole.  From  §  81  the  equa- 
tion of  a  conic,  when  the  axis  of  x  is  an  axis  of  the  conic  and  the 
axis  of  3/  is  a  directrix,  is 

{x  —  cf  +  y^  =  e^x^. 

We  may  transfer  to  new  axes  having  the  origin  as  the  focus  and 
the  axis  of  x  as  the  axis  of  the  conic  by  placing 

x  =  c  +  x',  y  =  y', 

thus  obtaining  x'^-\-  y'^  =  e'^{x'  +  cf. 

If  we  now  take  a  system  of  polar  coordinates  having  the  focus 
as  the  pole  and  the  axis  of  the  conic  as  the  initial  line,  we  have 

x'  =  r  cos  6,  y'  —  "f  sin  6. 


344 


POLAR  COORDINATES 


The  equation  then  becomes 

r^  =  e^  {r  cos  6  +  cf, 
which  is  equivalent  to  the  two  equations 


.  r  = 


1  —  e  cos  0 

ce 


1  +  e  cos  6 


Either  of  these  two  equations  alone  will  give  the  entire  conic. 
To  see  this,  place  6  =  6^  in  the  second  equation,  obtaining 


T,  = 


—  ce 


l-\-  e  cos  6^ 

Now  place  6  =  'Tr  -\-  6^  in  the  first  equation,  obtaining  r  =  —  r^. 
The  points  {9^,  rj  and  (tt  +  6^,  —  r^  are  the  same.  Hence  any 
point  which  can  be  found  from  the  second  equation  can  be  found 
from  the  first. 

ce 


Therefore 


r  = 


e  cos 


d 


is  the  required  polar  equation. 

186.   Examples.    We    shall   now   give    examples   of    the   use 

of  polar  coordinates    in   solving 
problems. 

Ex.  1.  Prove  that  if  a  secant  is 
drawn  through  the  focus  of  a  conic, 
the  sum  of  the  reciprocals  of  the  seg- 
ments made  by  the  focus  is  constant. 

Let  PiPo  (fig-  208)  be  any  secant 
through  the  focus  F,  and  let  FPi  =  ri 
and  FPi  =  r^,  and  the  angle  MFP  =  0. 
Then  the  polar  coordinates  of  Pi  are 
(ri,  0)  and  those  of  Pa  are  (ra,  d  +  ir). 
From  the  polar  equation  of  the  conic 
we  have 


Fig.  208 


ri  = 


r2 


1  —  ecos^ 
ce 


Hence 


1- 
Ti      r^      ce 


e  cos  {d  +  tt) 


ce 

1  +  e  cos  0 


DIRECTION  OF  A  CUEVE 


345 


Ex.  2.  Find  the  locus  of  the  middle  points  of  a  system  of  chords  of  a  circle 
all  of  which  pass  through  a  fixed  point. 

Take  any  circle  with  the  center  C  (fig.  209)  and  let  0  be  any  point  in  the 
plane.    If  O  is  taken  for  the  pole  and  OC  for  the  initial  line  of  a  system  of  polar 

coordinates,  the  equation  of  the  ^^^^  p 

circle  is 

r2-2r6cose  +  62_a2  =  0.  (1) 

Let  P1P2  be  any  chord  through 
0  and  let  OPi  =  ri,  OP2  =  rg. 
Then  ri  and  r%  are  the  two  roots 
of  equation  (1)  which  correspond 
to  the  same  value  of  d.    Hence 

n  +  r2  =  2  6  cos  6. 

If  Q  is  the  middle  point  of 
P1P2,  and  we  now  place  OQ  =  r, 
we  have 

ri  +  r2      ,         . 

r  = =  ocos^.  ,,       „^^ 

2  iiG.  209 

But  this  is  the  polar  equation  of  a  circle  through  the  points  0  and  C. 

187.  Direction  of  a  curve.  The  direction  of  a  curve  expressed 
in  polar  coordinates  is  usually  determined  by  means  of  the  angle 
between  the  tangent  and  the  radius  vector.    Let  F(r,  6)  (fig.  210) 

be  any  point  on  the  curve, 
VQ       /'■  PT  the  tangent  at  P,  and 

i/r  the  angle  made  by  PT 
and  the  radius  vector  OP. 
Give  6  an  increment 
A6=P0Q,  expressed  in 
circular  measure,  thus  fix- 
ing a  second  point  of  the 
curve  Q(r  +  /^r,  6  +  A6). 
To  determine  Ar  describe 
a  circle  with  center  0  and  radius  OQ,  intersecting  OP  produced 

in  R.    Then 

OB  =  OQ  =  r  +  Ar, 

PE  =  Ar, 

and  SiTG  PQ  =  As, 


Fig.  210 


s  being  measured  from  some  initial  point  A. 


346  POLAR  COORDINATES 

Draw  also  the  chord  PQ  and  the  straight  line  QS  perpendicular 
to  OP  and  meeting  it  in  S.    Then 

^^  =  (r  +  Ar)  sin  A^, 
OS  =  {r  +  At)  cos  A6, 
SB  =  OE-OS 

=  (r  + Ar)(l— cosA^), 
and  PS  =  PE-SB 

=  Ar  —  {r  +  Ar)  (1  —  cos  A^). 

As  A0  approaches  zero,  the  chord  PQ  approaches  the  limiting 
position  PT  and  the  angle  EPQ  approaches  i/r.  But  in  the 
triangle  SPQ 

SQ 


tanEPQ  =  ^^ 


(r  +  Ar)  sin  A^ 


Ar  —  (r  +  A?')  (1  —  cos  A^) 
sin  A^ 


(r  +  Ar) 


A6> 


Ar      ,         .    ,  1  —  cos  A^ 
-—  —  (r  +  Ar)  - 


(1) 


A^     '  '       Ad 

Now  as.:A^  approaches  zero 

T-     /        A   \  T-     sinA^      . 

lam  (r  +  A?-)  =  r,  Lim  — —r-  =  1, 

A0 

,.     Ar      ch  ,      ,.      1— cosA^      a/ciki\ 

Hence,  by" taking  the  limit  of  (1), 

■tan '«ir  =  4^.  (2) 

ar 

•To 

If  it  is  desired  to  find  the  angle  MNP  =  (f),  it  may  be  done  by 
the  evident  relation 


DERIVATIVES  WITH  RESPECT  TO  THE  ARC     347 

188.  Derivatives  with  respect  to  the  arc.    In  the  triangle  FQS 
(fig.  210) 

SQ 


BmSPQ- 


chord  P^ 
SQ         QxcPQ 


arcFQ  chord  P^ 

_(r  +  Ar)  sin  A^     arc  PQ 
As  chord  P^ 

^   ,sinA(9  Ad     arc  PQ 
=  (r  +  Ar) 


A0      As   chord  Pg  , 

As  A^  approaches  zero,  SPQ  approaches  yjr,  Lim  — —^  =  1,  and 

Lim  f^'^f^    =  1  (§  104) ;  hence 
chord  P^         ^^        ^ 

sin '\lr  =  r—--  (1) 

as 


By  dividing  (1)  just  obtained  by  (2)  of  the  previous  article, 

dr 
ds 


cos  ■\(r  =  — —  (2) 


From  (1)  and  (2)  we  obtain 


/dsY 
By  multiplying  (3)  by  |  -r^  I  we  obtain 


d0j    ^\ddi       '        ^ ' 


/dsV 
and  by  multiplying  (3)  by  ( ^r- )  we  obtain 


CI 

drj      \    dr 


'J.yjr'^)'+l.  (6) 


348 


POLAR  COORDINATES 


189.  Area.     Let  C  (fig.  211)  be  a  fixed  point  and  P  {r,6)  a 
variable  point  on  the  curve  r  =f(6),  and  let  A  denote  the  area 

of  the  figure  OCF,  bounded  by 
the  arc  of  the  curve  CP  and 
the  radii  OC  and  OP.  Then  A 
is  a  function  of  6,  since  the 
value  of  0  fixes  the  position  of 
the  point  P.  If  6  is  increased 
by  A^  =  angle  POQ,  A  is  in- 
creased by  A  A  =  area  POQ. 
From  0  describe  arcs  of  circles 
PS  and  QR  with  radii  OP  =  r 
and  OQ  =  r  +  Ar  respectively. 
Yio.  211  Then  in  the  figure 

aresi POS  <AA<  aresi EOQ. 

But  the  area  of  tlie  sector  of  the  circle  POS  is 

lOP-PS=lr'Ae, 
and        SiTeaBOQ=^^OQ-BQ==^^{r  +  AryAe. 
We  have  then       ^  r^AO  <AA<l(r  +  ArfAd ; 

AA 
whence  ^t^  <  -rz  <\{^  +  Ar)l 


Taking  now  the  limit  as  A^  approaches  zero,  we  have 

Ex.    Find  the  area  of  a  loop  of  the  lemniscate  r^  =  2  a*  cos  2^. 

We  will  take  C  as  the  point  for  which  ^  =  0,  and  P  as  any  point  for  which 


0<»<-- 
4 

Then 

dA 

=  o2  cos  2  »  ; 

de 

whence 

0^ 

^  =  —  sin  2  ^  +  c. 

But  when  e  =  0,  A  =  0;  therefore  c  =  0.     Also  when  6  =  -,  A-  I  area  of 

^  4 

the  loop.    Hence  the  area  of  the  loop  is  a^  sin  -  =  a^. 


PROBLEMS  349 

PROBLEMS 


Plot  the  following  curves : 


1.  r=asin2^.  13.  r  =  a(l  +  cos2^). 

2.  r=acos3tf.  14.  r  =  a(l  +  2  cos2e). 

3.  r  =  atan^.  15.  r  =  a(l— cos2fl). 

4.  r  =  a(l  +  sintf).  16.  r  =  a(l  +  cosSfl). 

5.  r  =  a(2  +  sin«).  17.  r  =  a(l  +  2  cos3tf). 

6.  r  =  a(l  +  2  sintf).  18.  r  =  4  +  5  cos  5^. 

7.  r  =  a0~i.  19.  r  =  2  +  sin  S  0. 


8. 

r  = 

asec2-. 
2 

9. 

r  = 

a 
0-b 

10. 

r  = 

a  —  he. 

11. 

r  = 

.  e 

a  sin  - . 

2 

12. 

r  = 

e 

■  a  cos  - . 
3 

20. 

r  =  a  tan  -  • 

2 

21. 

r  =  a  sin8  - . 
■     3 

22. 

r2  =  a2  sin  0. 

23. 

r2  =  a^sinStf. 

24. 

r  cos  5  =  a  cos  2  tf. 

25. 

a           a 

r  = h  ^ 

cos  0      sm  fl 

Find  the  points  of  intersection  of  the  following  pairs  of  curves : 

26.  r  cos  (  0 ]=  a,  r  cos  10 i  =  a. 

27.  r cos /  ^ 1  =  — ,  r  =  asin0. 

\        2;        4 

28.  r2  =  a2  sin  0,  r^  =  a^  sin  3  0. 

29.  r  =  a  sin  20,  r  =  a (1  —  cos  2 0).  [(ri,  ^1)  and  ( —  n,  ^i  +  tt)  are  the  same 
points.] 

30.  0  is  a  fixed  point  and  LK  a  fixed  straight  line.  If  any  line  through  O 
intersects  LK  in  Q  and  a  point  P  is  taken  on  this  line  so  that  OP  •  OQ  =  k^, 
find  the  locus  of  P. 

31.  A  straight  line  OA  of  constant  length  revolves  about  0.  From  4 
a  pei-pendicular  is  drawn  to  a  fixed  straight  line  through  0,  intersecting  it 
in  B.  From  B  a  perpendicular  is  drawn  to  OA  intersecting  it  in  P.  Find  the 
locus  of  P. 

32.  MN  is  a  straight  line  perpendicular  to  the  initial  line  at  a  distance  a 
from  0.  From  0  a  straight  line  is  drawn  to  any  point  B  of  MN.  From  B  a 
straight  line  is  drawn  perpendicular  to  OB.  intersecting  the  initial  line  at  C. 
From  C  a  line  is  drawn  perpendicular  to  BC,  intersecting  MN  at  D.  Finally, 
from  I>  a  straight  line  is  drawn  perpendicular  to  CD,  intersecting  OB  at  P. 
Find  the  locus  of  P. 


350  POLAR  COORDINATES 

Transform  the  following  equations  to  polar  coordinates  : 

33,  y2  =  4px,  36.  x2  +  2/2  -  8  ox  -  8  ay  =  0. 

34.  xy  =  7.  37.  x*  +  x2y2  _  a2y2  =  q. 

35   ^  +  ?^  =  1  ^^-  ^'^^  +  ^')'  =  '''  (•^'  ~  ^'>- 

'  a2      62       •  39.  x^  +  y3-3axy^0. 

40.  Find  the  polar  equation  of  the  cissoid  when  the  pole  is  A  and  the  initial 
line  is  OA  (fig.  91). 

41.  Find  the  polar  equation  of  the  strophoid  (1)  when  the  pole  is  0  and 
the  initial  line  OA  (fig.  92);  (2)  when  the  pole  is  A  and  the  initial  line  is  OA. 

42.  In  the  strophoid  (fig.  92)  show  that 

112 
AP.APi  =  a:^,     and    -t^  +  -!—  =  -^, 
AP      APi      AN 

where  AN  is  the  projection  of  ^0  on  AD. 

Transform  the  following  equations  to  rectangular  coordinates : 

43.  rcosM-- J +  rcosM+-|  =  12.     46.  r  =  atan«. 

44.  r  =  asin«.  47.  r2  =  a2sin«. 

g 

45.  r  =  a(cos2^  +  sin2«).  48.  r2  =  a2sin-. 

49.  Find  the  Cartesian  equation  of  the  rose  of  four  petals  »•  =  a  sin  2  6. 

50.  Find  the  Cartesian  equation  of  the  cardioid  r  =  a(l  —  cos 5). 

51.  Find  the  Cartesian  equation  of  the  ovals  of  Cassini 

r*  -  2a2r2cos2^+ a*  -  6*  =  0. 

52.  Find  the  Cartesian  equation  of  the  limagon  r  =  a  cosO  +  b. 

53.  Find  the  Cartesian  equation  of  the  conchoid  r  =  a  sec  5  +  6. 

54.  Find  the  Cartesian  equation  of  the  logarithmic  spiral  r  =  e"^. 

55.  In  a  parabola  prove  that  the  length  of  a  focal  chord  which  makes  an 
angle  of  30°  with  the  axis  of  the  curve  is  four  times  the  focal  chord  perpen- 
dicular to  the  axis. 

^    ,  56.  A  comet  is  moving  in  a  parabolic  orbit  around  the  sun  at  the  focus  of 
,the  parabola.    When  the  comet  is  100,000,000  miles  from  the  sun  the  radius 
.  vector  makes  an  angle  of  60°  with  the  axis  of  the  orbit.    What  is  the  equation 
of  the  comet's  orbit  ?    How  near  does  it  come  to  the  sun  ? 

57.  A  comet  moving  in  a  parabolic  orbit  around  the  sun  is  observed  at  two 
■  Boinfs  of  its  path,  its  focal  distances  being  5  and  15  million  miles  and  the  angle 

between  them  being  90°.   What  is  its  distance  from  the  sun  when  it  is  nearest  it  ? 

58.  If  a  straight  line  drawn  through  the  focus  of  an  hyperbola,  parallel  to 
an  asymptote,  meets  the  curve  at  P,  prove  that  FP  is  one  fourth  the  chord 
through  the  focus  perpendicular  to  the  transverse  axis. 


PROBLEMS  351 

59.  The  focal  radii  of  a  parabola  are  extended  beyond  the  curve  until  their 
lengths  are  doubled.    Find  the  equation  of  the  locus  of  their  extremities. 

60.  If  Pi  and  P2  are  the  points  of  intersection  of  a  straight  line  drawn  from 
any  point  O  to  a  circle,  prove  that  OPi  ■  OP2  is  constant. 

61.  If  Pi  and  P^  are  the  points  of  intersection  of  a  straight  line  from  any 
point  O  to  a  fixed  circle,  and  Q  is  a  point  on  the  same  straight  line  such  that 

00  =  ^  ^^^ '  ^^^ ,  find  the  locus  of  Q. 
^      OP1  +  OP2  ^ 

62.  Secant  lines  of  a  circle  are  drawn  from  the  same  point  on  the  circle, 
and  on  each  secant  a  point  is  taken  outside  the  circle  at  a  distance  equal  to  the 
portion  of  the  secant  included  in  the  circle.    Find  the  locus  of  these  points. 

63.  From  a  point  0  a  straight  line  is  drawn  intersecting  a  fixed  circle  at  P, 
and  on  this  line  a  point  Q  is  taken  so  that  OP  ■  OQ  =  k^.    Find  the  locus  of  Q. 

64.  Find  the  polar  equation  of  a  conic  if  the  pole  is  a  vertex  and  the  initial 
line  an  axis. 

65.  Find  the  locus  of  the  middle  points  of  the  focal  chords  of  a  conic. 

66.  Find  the  locus  of  the  middle  points  of  the  focal  radii  of  a  conic. 

67.  If  P1PP2  and  QiFQ-2  are  two  perpendicular  focal  chords  of  a  conic, 

prove  that 1 is  constant. 

PiF-FPi.    Q1F.FQ2 

68.  Prove  that  the  angle  between  the  normal  and  the  radius  vector  to  any 
point  of  the  lemniscate  is  twice  the  angle  made  by  the  radius  vector  and  the 
initial  line. 

69.  Show  that  for  any  curve  in  polar  coordinates  the  maximum  and  the 
minimum  values  of  r  occur  in  general  when  the  radius  vector  is  perpendicular 
to  the  tangent. 

70.  If  a  straight  line  drawn  through  the  pole  0  perpendicular  to  a  radius 

t" 
vector  OP  meets  the  tangent  in  A  and  the  normal  in  B,  show  that  OA  =  — 

dr  ^^ 

and  OB  =  —  v; 

de  ,  de 

These  are  called  the  polar  subtangent  and  the  polar  subnormal  respectively. 

71.  If  p  is  the  pei-pendicular  distance  of  a  tangent  from  the  pole,  prove 
that  p  = 


nRI 


72.  When  a  point  traverses  the  curve  r=f{d)  with  a  uniform  angular 
velocity,  find  the  rate  at  which  r  is  changing  and  the  rate  of  the  point  along 
the  curve. 

73.  When  a  point  moves  along  the  curve  r=/{S)  at  a  uniform  rate,  find 
the  rates  at  which  r  and  6  are  changing. 


352  POLAR  COOEDINATES 

74.  Find  the  velocity  of  a  point  moving  in  a  lima^on  when  B  changes 
uniformly. 

75.  A  point  moves  along  the  radius  vector  with  a  constant  velocity  a,  while 
the  radius  vector  revolves  about  0  with  a  constant  velocity  w.  Find  the  path  of 
the  point. 

76.  Find  the  total  area  bounded  by  the  curve  r^  =  a^  sin  0. 
11.  Find  the  area  of  a  loop  of  the  curve  r^  =  a^  sin  3  d. 

78.  Find  the  area  swept  over  by  the  radius  vector  of  the  spiral  of  Archimedes 
as  0  changes  from  0  to  ir. 

79.  Find  the  area  swept  over  by  the  radius  vector  of  the  logarithmic  spiral 
as  0  changes  from  0  to  tt. 

Q 

80.  Find  the  area  swept  over  by  the  I'adius  v-ector  of  the  curve  r  =  asin- 
as  0  changes  from  0  to  2  tt. 

81.  Find  the  area  swept  over  by  the  radius  vector  of  the  curve  r  =  atanfl 

TT 

as  0  changes  from  0  to  —  • 
4 

82.  Find  the  total  area  of  the  limagon  (6  >  a). 

83.  Find  the  total  length  of  the  cardioid. 

84.  Prove  that  the  length  of  an  arc  of  the  logarithmic  spiral  is  proportional 
to  the  difference  of  the  radii  vectores  drawn  to  its  ends. 

85.  Show  that  if  the  angle  between  the  tangent  to  a  curve  and  the  radius 
vector  to  the  point  of  contact  is  one  half  the  vectorial  angle,  the  curve  is  a 
cardioid. 


CHAPTER  XVI 


CURVATURE 


190.  Definition  of  curvature.  If  a  point  describes  a  curve  the 
change  of  direction  of  its  motion  may  be  measured  by  the  change 
of  the  angle  <f>  (§  59). 

For  example,  in  the  curve 
of  fig.  212,  a  AP^  =  s  and 
F^I^  =  As,  and  if  ^^  and  (f)^ 
are  the  values  of  <f)  for  the 
points  i^  and  i^  respectively, 
then  ^2  ~"  ^1  i^  the  total 
change  of  direction  of  the 
curve  between  ij  and  I^. 
If  ^2  —  ^^  =  A<^,  expressed 
in  circular  measure,  the 
A^ 
As 


ratio 


is    the    average 


Fig.  212 


change  of  direction  per  linear  unit  of  the  arc  -^^.    Regarding 

^  as  a  function  of  s  and  taking  the  limit  of  — -^  as  As  approaches 

zero  as  a  limit,  we  have  -^  >  which 
as 

is    called    the    curvature   of   the 

curve  at  the  point  Py    Hence  the 

curvature  of  a  curve  is  the  rate 

of  change  of  the  direction  of  the 

curve  with  respect   to   the  length 

of  the  arc  (§  109). 

If  -~  is  constant,  the  curvature 
as 

is  constant   or  uniform;    other- 
Applying  this  definition  to  the 


Fig.  213 

wise  the  curvature  is  variable. 


circle    of    fig.  213    of    which    the    center    is   C   and   the    radius 


353 


354 


CURVATUKE 


Hence 


-J-  =  -,  and  the  circle  is  a  curve  of  constant 
ds      a 


is  a,  we   have   A^  =  -?^Ci^ ;    and  hence   As  =  aA<j).     Therefore 

A^_l 

As      a 

curvature  equal  to  the  reciprocal  of  its  radius. 

191.  Radius  of  curvature.  The  reciprocal  of  the  curvature  is 
called  the  radius  of  curvature,  and  will  be  denoted  by  p.  Through 
every  point  of  a  curve  we  may  pass  a  circle,  %vith  its  radius  equal 
to  p,  which  shall  have  the  same  tangent  as  the  curve  at  the  point, 
and  shall  lie  on  the  same  side  of  the  tangent.  Since  the  curva- 
ture of  a  circle  is  uniform  and  equal  to  the  reciprocal  of  its 
radius,  the  curvatures  of  the  curve  and  the  circle  are  the  same, 
and  the  circle  shows  the  curvature  of  the  curve  in  a  manner 
similar  to  that  in  which  the  tangent  shows  the  direction  of  the 
curve.    The  circle  is  called,  the  circle  of  curvature. 


Since  the  curvature  is  -^> 
ds 


_}__ds_ 
^^  d4~  d4>' 
ds 


(by  (6),  §  96) 


K  the  equation  of  the  curve  is  in  rectangular  coordinates. 


and 


whence 


dx      N        \dxl 

(by  §  105) 

\dx/ 

(by  §  59) 

A 

d<^           dx^ 

dx      ^Jdy^ 
\dx) 

ds 

ds       dx 
'  ^^d4)~'df 

(by  (8),  §  96) 

dx 

Mm 

i 

d'y 

cb? 

RADIUS  OF  CURVATURE  355 

In  the  above  expression  for  p  there  is  an  apparent  ambiguity  of 
sign,  on  account  of  the  radical  sign.  If  only  the  numerical  value 
of  p  is  required,  a  negative  sign  may  be  disregarded. 

Ex,  Find  the  radius  of  curvature  of  the  ellipse h  —  =  1. 

Here  dy_^_^ 

dx         a?y 

and  — ^  = 

dx2  a^y^ 

{a*y'^  +  6*x2)* 
•■•  P  = 


Another  formula  for  p,  i.e, 


r      /dx^^-^^ 


(Tx 

df 

may  be  found  by  defining  ^  as  the  angle  between  OY  and  the 
tangent,  and  interchanging  x  and  y  in  the  above  derivation. 

ds 
192.  According  to  the  definition,  i.e.  p  =  -tt'  it  is  evident  that 

d<p 

p  is  positive  when  s  is  measured  so  that  s  and  </>  increase  at  the 
same  time,  and  is  negative  when  one  increases  as  the  other  decreases. 
For  convenience  we  shall  assume  in  the  following  work  that  s 
always  increases  from  left  to  right  *  along  the  curve  (figs.  214-217). 
Then  ^  is  always  in  the  first  or  the  fourth  quadrant,  and  hence 
sec  ^  is  always  positive. 

But  sec  </>  =  Vl+tan^^=  \l  1  +  ( -^^ )  •    Therefore  in  the  formula 

NIII    . 

da? 

the  sign  of  p  is  the  same  as  the  sign  of  -—•    Hence  p  is  positive 

when  the  curve  is  concave  upward,  and  negative  when  the  curve 
is  concave  downward. 

•The  results  and  the  proof  are  the  same  if  s  is  measured  from  right  to  left  along 
the  curve ;  hence  the  proof  is  left  to  the  student. 


356 


CURVATURE 


193.  Coordinates  of  center  of  curvature.  The  center  of  the 
circle  described  in  §  191  is  called  the  center  of  curvature  corre- 
sponding to  the  point.  Let  C{a,  /3)  (fig.  214)  be  the  center  of 
curvature  corresponding  to  the  point  P{x,  y)  of  the  curve.  Draw 
CL  and  FM  parallel  to  OY,  and  NR  through  P  parallel  to  OX. 
Then 

OL  =  0M+  ML  =  OM+  PiV; 


Now 
and 


LC  =  LN+NC  =  MP  +  NC. 
ZBPC  =  <f>+ 90°, 
PC  =  p, 


Fig.  214 


since  p>0,  the  curve  being  concave  upward.    Therefore,  by  the 
definition  of  the  trigonometric  functions, 

PiV  =  PC  cos  i^PC  =  /?  cos  (<^  +  90°)  =  - /3  sin  </>, 
NC  =  PC  sin  PPC  =  p  sin(<^  +  90°)  =  p  cos<^. 
.'.  a  =  X  —  p  sin<f>, 
^  =  y  i-  pcos<f). 


There  are  three  other  cases  represented  in  figs.  215,  216,  217 
respectively.  The  construction  in  all  these  figures  is  the  same 
as  in  fig.  214,  and  the  proof  from  fig.  215  is  the  same  as  that 


EVOLUTE  AND  INVOLUTE 


357 


just  given.    The  proof  from  figs.  216  and  217  differs  only  in  that 
RFC  =  0—90°,  and  PC  ~  —  p,  since  /a  <  0,  the  curve  being  con- 
cave   downward.     Hence    the 
above    expressions    for  a  and 
/3  are  universally  true. 

Since   cos  ^  = 


and 


sin0  = 


dx 


Fig.  210 


the  formulas  for  a  and  /S  may  be  written 


a  =  x 


dx\_        \dx/  J 


^  =  y  + 


v< 


c£y_ 
dx^ 
dy^^' 

dx, 


dx^ 


Fio.  217 


In  the  example  of  §  191  we  found 
dx  a'hj 


Ex.  Find  the   coordinates   of 
tlie  center  of  curvature  for  any 

x^      w2 
point  of  the  ellipse  — | =  1. 

a'2      62 


and     -^—  = 


Substituting  in  the  above  formulas  and  simplifying,  we  have 
/a2  -  62\  /a2  _  62N, 

"  =  (-^r-)-^      ^  =  -(-6^)^'- 

194.  Evolute  and  involute.    With  the  single  exception  when 

d^v 

— ^  =  0,  in  which  case  p  becomes  infinite,  there  will  be  a  center 

d3(r 

of  curvature  corresponding  to  each  point  of  the  curve.    The  locus 


358 


CURVATURE 


b* 


y", 


X2  y2 

—  +  ~  =  1. 

a-2      62 


of  these  centers   of  curvature  is   a  curve  called  the  evolute  of 
the  given  curve,  and  the  given  curve  is  called  the  involute.    In 

fig.  218  (1)  is  the  involute 
and  (2)  is  the  evolute. 
To  find  the  evolute  we 
find  the  coordinates  of 
the  center  of  curvature 
in  terms  of  x  and  y,  and 
then  eliminate  x  and  y 
from  these  two  equations 
by  the  aid  of  the  equa- 
tion of  the  curve. 

Ex.  To  find  the  evolute  of 
the    ellipse,    we    have    then, 
in  the   example  of    the  last 
article,  to  eliminate  x  and  y  from  the  three  equations 

a2-62   -  „  a2-62 

a  = -— a:%        /3  = - 

a* 

From  the  first  two  equations 

X  _  /    aa    \ 
o  ~  W  -  62/ ' 

b  W  -  by 

Substituting  these  values  in  the 
third  equation  and  simplifying,  we 
have 

{aa)i  +  (6^)^  =  (o2  -  62)? 

as  the  equation  of  the  evolute.  The 
ellipse  and  its  evolute  are  shown  in 
fig.  219. 

It  may  be  noted  that  equa- 
tions expressing  a  and  y3  are, 
in  fact,  the  parametric  repre- 
sentation of  the  evolute,  x  and 
y  being  two  independent  param- 
eters connected  by  the  equation 
of  the  given  curve. 


Fig.  219 


EVOLUTE  AND  INVOLUTE  359 

195.  Properties  of  evolute  and  involute.  From  the  equations 
a  =  x  —  p  sill  4>,  /3  =  y  +  /)  cos  </>,  we  may  find  the  slope  of  the 
evolute  at  any  point  by  assuming  a,  /3,  x,y,p,  and  </>  as  functions 
of  s,  the  length  of  arc  along  the  involute.    Then 


da  _dx  d^       •    M^P 

ds      ds  ds  ds 

=  cos  (f)  —  p  COS  <^  ( -  )  —  sin  <^ 

=  — sinrf)-f-- 
ds 

dfi     dy  .     .  d(b  ,  ,  dp 

-7-  =  -/^— /^sm</>-^  +  cos</>-p 
as      ds  ds  ds 


dp 
ds 


=  sbi(f)  —  p  sin (f> {-)-{-  cos ^ -~ 
\p/  ds 

=  cos<i— • 
as 

^  d£ 

.    ds  ^     ,      ,    ^    ds       dB 

..-—  =  —  ctn  d> ;    but    -—-  =  -—> 
da  da      da 

ds  ds 

by  (8),  §  96 ;  and  if  tan^'  is  the  slope  of  the  evolute  at  the  assumed 

dfi 
point,  —  =  tan  </>',  and  hence  tan  <f)'  =  —  ctn  (f).     Hence  <f>'  and  </> 

differ  by  90°,  and  the  tangent  to  the  evolute  at  any  point  is 
perpendicular  to  the  tangent  to  the  involute  at  the  corresponding 
point  (fig.  218). 

If  we  square  and  add  the  above  equations,  we  have 


/da\ 
\ds) 


daV    (d^Y_(dp 
ds/      \ds/      \ds 


But  if  we  denote  the  length  of  arc  along  the  evolute  by  s',  we 

have  _=  -   i_|_/z^\  ;  and  if  we  regard  s',  a,  ^,  as  expressed  in 
da       N        \da) 

terms  of  s,  the  length  of  arc  along  the  involute,  we  have 


360  CURVATURE 


ds  I         \ds/  . 

da  ~~      I         /daV ' 

ds     n|       W 


whence 
Hence 


as        ^\ds/      \ds/ 
c?s/         \ds/ 


and  -y-  =  ±  , 

as  as 


ds'  _      C?/3 

6?S 

•'•  s'  =  ±p  +  c.  (by  §  110) 

/if  follows,  then,  that  as  the  center  of  curvature  moves  along  the 
evolute  the  radius  of  curvature  increases  or  decreases  hy  exactly  the 
distance  traversed  hy  the  center  (fig.  218). 

From  these  two  properties  we  see  that  an  involute  may  be 
described  by  a  pencil  attached  to  the  end  of  a  string  which  is 
unwound  from  the  evolute,  the  free  portion  being  kept  taut  and 
tangent  to  the  evolute.  From  any  one  evolute  any  number  of 
involutes  may  be  described  by  changing  the  length  of  the  string. 

196.  Radius  of  curvature  in  parametric  representation.  If  x 
and  y  are  expressed  in  terms  of  any  parameter  t,  the  radius  of 
curvature  may  be  fovmd  as  follows: 


ds       dt 


But 

and 

cix 

dt 


p  = 

d(f)      d<f) 

dt 

ds 
dt' 

-4m<tT 

dy 

<!>- 

=  tan-^-/  =  tan-^--; 
dx              ax 

(by  (8),  §  96) 


RADIUS  OF  CURVATURE 


361 


whence 


dt 


dx\  dj^y      (dy\  d?x 
di/d^~\di)df 


ldy\ 

\dti 


(dxY 
\dtl 


dx  d^y      dy  d^x 
Tt"df~^t'~df 


(dxV     /dy 
\dt)      \dt 


Therefore,  by  substitution, 


P  = 


dx  d^y     dy  d?x 
'dt'~de~~dt"de 


Ex.  Pind  the  radius  of  curvature  of  the  cycloid 


Here  the  parameter  is  0 


x  =  a<t> 

—  a  sin  0, 

y  =  a- 

acos<f>. 

dx 
d<t> 

acoscft, 

d^x 

—  =  asinc^, 

dy 

-^  =  asin^, 

d(p 

d"'y 

Hence,  by  substitution,      p 


=  a  cos<t>. 

_  [a2(l-cos0)2  +  a2.sinV]^ 

a  (1  —  cos  <p)  ■  a  cos  <p  —  a  sin  <p  {a  sin  <p) 
=  -2^a(l-cos0)^ 


=  -2^a-(2sin^^\ 


—  4a  sin 


197.  Radius  of  curvature  in  polar  coordinates.  The  equation 
of  any  curve  in  polar  coordinates  may  always,  theoretically,  be 
expressed  in  the  form  r=f{d).    Then,  since  r  may  be  regarded 


362  CURVATURE 

as  a  function  of  6,  the  equations  x  =  r  cos  B,  y  —  r  sin  Q,  are  the 
parametric  equations  of  a  curve.  From  them  we  may  accordingly 
derive  the  formula  for  p  in  polar  coordinates  by  substituting  in 
the  formula  of  §  196  as  follows: 

dx      dr        n  •    /I 

dy      dr    .    a  ,  n 

J  =  -^smfl  +  .cos«, 

dr  V      drT  dv 

_|  =  _siu^  +  2^cos(>-.sin«. 

Substitutmg  these  values   and    simplifying,  we   have,  as   the 
required  formula, 


\ddl  d&' 

Ex.  Find  the  radius  of  curvature  of  the  cardioid  r  =  a(l  —  cos^). 

Here  —  =  a  sin  d  and  —  —  a  cos  6. 

de  dff^ 

_  [a2(l  -  cos^)2  +  a?  sin2fl]i 

'  a2(l-  cos^)2  +  2a2sin2tf  _  a(l  -  cose)acos9 

[2a2(l-cos^)]3      2?a„  „,i 

= ^^ -^  = (1  -  costf)*, 

02(3-3  00861)  3    ^ 

p  =  f(2ar)i. 

PROBLEMS 

fj      -         _  - 

1.  Find  the  radius  of  curvature  of  tlie  catenary  y  —  ~  {e"  +  e   "). 

2.  Find  the  radius  of  curvature  of  the  cissoid  y^  — 


2a  —  X 

3.  Find  the  radius  of  curvature  of  the  four-cusped  hypocycloid  x^  +  y^  =  a*. 

4.  Find  the  radii  of  curvature  of  the  curve  a*y^  =  a^*  —  x^  at  the  points 
(0,  0)  and  (a,  0). 

5.  Find    tlie    radius   of    curvature    of    the    curve  (-)  +(-)   =1  at    the 
point  (0,  b).  ^"^       ^^^ 


PROBLEMS  363 

6.  Find  the  radii  of  curvature  of  tlie  curve  y^  =  ax{x  —  Sa)  at  the  points 
where  it  crosses  the  axis  of  x. 

7.  Find  tlie  radius  of  curvature  of  the  curve  e'^  =  sin  y  at  the  point  (xi,  yi). 

8.  Find  the  slope  and  the  radius  of  curvature  of  the  curve  y  +  log(l  —  x^)  =  0 
at  the  origin  of  coordinates. 

9.  Show  that  the  radius  of  curvature  of  the  curve  r  =  a  (sin  6  +  cos  0)  is 
constant. 

10.  Find  the  radius  of  cui^ature  of  tlie  curve  r  =  a  (2  cos^  —  1). 

11.  Find  tlie  radius  of  curvature  of  the  curve  r  =  a  sin^  - .    Find  the  greatest 
and  the  least  values  of  the  radius  of  curvature. 

12.  Find  the  radius  of  curvature  of  the  lemniscate  r^  =  2a^  cos  2  8. 

13.  Given  the  curve  x  =  2  cos  t  —  cos  2t,  y  =  2  sint  —  sm2 1.   Find  the  radius 
of  curvature  in  terms  of  t,  and  show  that  it  will  be  greatest  when  t  =  Tr. 

14.  Find  the  e volute  of  the  parabola  y^  =  ipx. 

15.  Find  the  radius  of  curvature  of  the  tractrix 


a,      a  +  Va2-x2         _ 

y  —  -log  — ■ Va2  -  x2. 

2       a  _  Va2  -  x2 

16.  Prove  that  the  evolute  of  the  tractrix  is  the  catenary. 

17.  Prove  that  the  evolute  of  a  cycloid  is  an  equal  cycloid. 

18.  Find  the  evolute  of  the  four-cusped  hypocycloid  x  =  a  cos^4>,  y  =  a  sin^^. 

19.  Find  the  evolute  of  the  ellipse  from  the  parametric  equations  x  =  a  cos^, 
y  =  bsm  (f>. 

20.  Prove  that  the  center  of  curvature  of  any  point  of  the  logarithmic  spiral 
is  the  point  of  intersection  of  the  normal  with  the  perpendicular  to  the  radius 
vector. 

21.  Find  the  circle  of  curvature  of  the  curve  y  =  er'^  when  x  =  0. 

22.  Show  that  the  catenary  ?/  =  ^(e^  +  e-^)  and  the  parabola  2/  =  1  +  ^x^ 
have  the  .same  tangent  and  the  same  circle  of  curvature  at  their  point  of 
intersection. 

23.  Find  the  point  of  minimum  curvature  on  the  curve  y  —  log  x. 

24.  Find  the  points  of  greatest  and  of  least  curvature  of  the  sine  curve 
y  =  sinx. 

25.  Find  the  points  on  the  ellipse  for  which  the  curvature  is  a  maximum  or 
a  minimum. 

26.  Show  that  the  curvature  of  the  parabola  y  =  ax?  +  6x  +  c  is  a  maximum 
at  the  vertex. 


364  CURVATURE 

27.  Find  the  condition  for  a  maximum  or  a  minimum  of  the  curvature  k, 
dx2 


where  A;  = 


Mm' 


28.  At  what  points  on  tlie  curve  y  =  log  sin  x  is  the  radius  of  curvature 
unity,  and  in  what  direction  from  the  point  on  tlie  curve  is  the  center  of 
curvature  ? 

29.  Show  that  the  product  of  the  radii  of  curvature  of  the  curve  y  —  ae~^ 
at  the  two  points  for  which  x  =  ±ais  a'^(e  +  e-i)^. 

30.  If  the  angle  between  the  radius  vector  to  the  point  of  contact  and  the 
straight  line  drawn  from  the  pole  perpendicular  to  the  tangent  is  either  a  maxi- 


ANSWEKS 

(The  answers  to  some  problems  are  intentionally  omitted.) 

CHAPTER  I 
Page  25 

1.  9.  3.  x-x2.  5.   17.  7.  1. 

2.  x-y.  4.  4.  6.  -18.  8.  2aJc 
9.  ahc  +  2fgh-aP-hg'^-ch'^.                   11.  Ix-Qy-b. 

10.  a62  +  hc^  +  ca^  -  ac^  -  ba^  -  cb^.  12.  3. 

13.  2aia2CiC2  +  01616202  +  02616301  —  a^c^  —  aScf  —  ai6,Jci  —  0261%. 

Page  26 

2^    13  zfcVei  33.  a;=  2,  2/=-2,  z  =  f. 

2        '  34.  a;  =  -  5,  2/  =  0,  z  =  4. 

25.  0,  6  ±  V39.  35.  X  =  -  1,  2/  =  0,  z  =  0. 

26.  8x  + 2/-13  =  0.  38.  x  =  i(a-26  + c  +  d), 

27.  x2  +  2/2  -  X  -  ?7  =  0.  7/  =  i  (a  +  6  -  2  c  +  d), 

28.  x2  +  2/2-3x  +  i/- 4  =  0.  z  =  i(«  +  ^  + c-2d), 

29.  x2  -  (a  +  6) X  +  tt6  -  /i-  =  0.  lo  =  { (-  2  a  +  6  +  c  +  d). 

30.  x3  -  (a  +  6  +  c)  x2  +  (a6  +  6c  39.  Xi :  X2  :  X3  =  3  :  -  5  :  -  2. 

+  ca  —  /2  —  </2  —  /i"-)  X  —  a6c  40.  xi :  X2  :  X3  =  1 :  —  2  :  3. 

-  2fgh  +  a/2  +  6^2  +  cA2  =  0.  41.  X3  =  0,  Xi :  X2  =  -  2  : 1. 

31.  X  =  1,  y  =  2.  42.  xi :  X2  :  X3  :  X4  =  4  :  —  3  :  2  :  5. 

32.  X  =  2,  y  =  -  1,  z  =  3.  43.  Xi  =  0,  X2  =  0,  X3  :  X4  =  3  :  2. 


51.  2/2 +  11 2/ +  12  =  1 
2  52.  2/  -  4  =  0. 

50.  2  2/2  +  6  2/  -  3  =  0.  53.  62  -  (a  +  c)2  =  0. 


CHAPTER   II 
Page  45 

1.   5  +  3  Vi.  7.  (-  I,  0).  10.  (-  1,  -  3i),  (1,  -  41). 

8.  (lA,  3H).  8.  (-  2|,  -  1|),  (-  If,  \l).     11.  (-  I,  If). 

6.  (H,  Hi)-  9-  (-5,  3  ±  V7).  12.  (-  i,  -  11). 

Page  46 

14.  (5,  -1).  16.  (-10,31).  18.   ^VSO,  1V53,  V26. 

15.  (-14,  17).  17.  (2,  1),  (4,  -  1).  19.  (15,  -3). 

366 


366 


A2TSWERS 


CHAPTER  III 

Page 

66 

21. 

tan-i|.              22. 

tau-if              28.  -. 
4 

Page 

67 

26. 

x-y  -4=0. 

32.   tan-iT-'j. 

27. 

8x  +  dy  +  83  =  0. 

33.  2x  +  8y-17  =  0. 

28. 

5x  -6y  =  0. 

34.  9x-Gy  -2  =  0. 

29. 

7x-2y  +  S  =  0. 

35.  x-y  +  2  =  0. 

30v 

3z-y-7  =  0. 

36.  4x-62/  +  16  =  0. 

81. 

X  +  1  =  0. 

37.  25x  +  loy-  24  = 

Page 

68 

44. 

2x  +  Sy  +  1  =  0. 

48.  IfVlS. 

45. 

i- 

^^    62  _  a2  _  aft 

46. 

(0,  2),  (0,  7),  (3,  5) 

;               V52  +  a2 

25.   5x  -  4j/  +  40  =  0. 


38.  5x-2y-10  =  0. 

39.  7x  +  4y +  28  =  0. 

41.  12x-1.3y-8  =  0. 

42.  3x-2y-7  =  0. 

43.  x+iy  -4  =  0. 


0. 


— ,  tan-13,  tan-15. 
4  ^' 


50. 


Vl3. 


47.  jJyVn, 

Page  69 


51.  6V2,  HV2. 


52.  |. 

58.  (3,  0). 

59.  2x  -  y-  4  =  0. 

60.  (-21,  -li). 

61.  (±fiVl3,±ffVi3). 


63.  (1,  1),  (- 3,  3).  _  66.  5x  +  2/-12  =  0,  x-5?/  +  8  =  0. 

64.  5x-j^-3  =  0;  2  V26.  67.  3x-4y  -2  =  0,  x  -  2  =  0. 

65.  17x  +  6j/  +  34  =  0,  x  +  18y  +  2  =  0. 


CHAPTER   IV 


Page 

94 

7. 

a  = 

Page 

95 

10. 

(1) 

11. 

(1) 

(3) 

12. 

(1) 

(2) 

(3) 

_  4  2 


8. 


2a 


26.  f, 


fc=|;  (2)A:>|;  (3)  A:<|. 

A;  =  0or4;  (2)  fc<0orA;>4; 

0<A;<4. 

A;  =  -  3  or  -  ^  ; 

-3<A;<-1; 

A;<  -  3  or  k>  —  |. 

|(-3±3  V-3). 


26.  2, -i, -1±V^,  {{1±- 

27.  0,  H±l±Vl3). 
0,  6,  ±  V^  ±  V^. 


28. 
29. 

30. 
31. 


0,  -2ffl,  —  (-l±5i). 


32.  x3  -  ax2  -  (a2  +  6)x  +  a^  -  ab 


±{a  +  1),  ±  (a  - 
6x3-  13x2  +  6x 
=  0. 


=  0. 


33.  x6  _  4aj;5  +  {4a2  -  62  -  26)x*  +  8  aSx^  -  (8  a26  -  263)x2  -  q. 

34.  x2  -  4  X  +  13^  0. 

35.  (2x  +  2-Vll)(2x+2  +VlT). 

36.  (2  X  +  3  +  V^)  (2  X  +  3  -  V-  2). 

37.  (2  ax  +  J  +4  V-  3)  (2  ax  +  J  -  ^  V^). 

38.  (X  +  a  +  Va)  (x  +  a  -  Va). 


ANSWERS  367 


89.  {ax  +  b  +  Va  +  62)  (ox  +  6  -  Va  +  ^). 

40.  (ax  +  b  +  i  Vb)  (ax  +  b  -  i  Vft). 

41.p2_29.  ^^    p^-2q  46.  p2-3r. 

42.  S^jg  —  p3.  ^2      '  47.  j)r. 

43.  -P.  46    ^'^-^g.  48.^. 

9  Q  r 

Page  96 

62.  1,  ^(3±v^).  66.  -3,  H-l±^^^)-  70--f,-l±'^. 

63.  2,2,  -1.  67.  ^,  §,  f.  71.   -  1,  _  1,  ±  i. 

64-  2,  2,  -  |.  68.  i,  I,  -  f    72.  3,  -  2,  f ,  -j. 

65.  -3,  f,  f.  _         69.  I,  -2dhV-2.  73.  2,  |,  1±V^, 

74.  4,  -  f ,  1(3  ±  V5).  80.  1,  -2,  -^,2  ±  Vs. 

75.  ±f,  J(3±V^>.  87.  1.41. 

76.  i,  -5,  1(1  J.V6).  88.    -1.52. 

77.  2,  3,  -  1,  -  1  ±  vCr2.  89.  2.05,  .59. 

78.  -2,  ±J,  |(_l±Vr3).  90.  1.18,2.87. 

79.  I,  ±  # ,  _  2  ±  Vs.  91.  .16,  2.93,  -  2.09. 


CHAPTER  V 
Page  118 

1.  12.  3.  0.  14.  (4|,  55f).         16.  lOf. 

2.  -3.        11.  32x  +  j/  +  45  =  0.        15.  tan-i^V         17.  (2,  9),  (- 2,  5). 

18.  4x  +  2/  +  2  =  0,  lOSx  +  27y  +  58  =  0. 

19.  (-1,  -6),  (1,  -512). 
Page  119 

20.  Increasing  if  x  >  —  2  ;  decreasing  if  x  <  —  2. 

21.  Increasing  if  x  <  0  or^  >  ^  ;  decreasing  if  0  <x  <  f. 

22.  Increasing  if  x  >  —  V2  ;  decreasing  if  x  <  —  V2. 

23.  Increasing  ifx>lor— l<x<0;  decreasing  if  0  <  x  <  1  or  x  <  —  1. 
24-   (I,  I)-  26.   (0,  i),  (±  2,  -  3|). 

26.  Maximum  value,  f  f  f  ;  minimum  value,  —  3. 

27.  Maximum  values,  —  12,  —  66  ;  minimum  values,  —  34,  —  88. 

29.  1  (a  +  6  -  Va2  +  6^  -  ab). 

30.  Altitude,  -;  base,  -• 

2  2 

31.  Altitude,  |  a  V3 ;  radius  of  base,  ^  a  Ve. 

32.  Altitude  is  one  third  the  altitude  of  the  cone.       83.   (1^,  f). 

84.  The  one  in  which  the  radius  of  the  circle  from  which  it  is  cut  is  one 
fourth  the  perimeter  of  the  sector. 

35.  Altitude  is  one  half  .a  side  of  the  base. 

37.   (1)  Height  of  the  rectangle  is  equal  to  the  radius  of  the  semicircle. 

(2)  Semicircle  of  radius—.  88.  J  a. 


368 

ANSWERS 

Page 

120 

39. 

Length  is  twice  the  breadth. 

52. 

-  2.21. 

40. 

.06  CIK 

53. 

2.09. 

44. 

Upward  if  x>  1; 

,    54. 

1.20,  3.13,  -1.38. 

downward  if  x  <  1. 

55. 

1.51,  -1.18. 

45. 

Upward  if  x  >  0, 
downward  if  x  <  0. 

56. 
57. 

2,  2,  -  3. 

1,  1,  -  a  ±  Va-^  -  a. 

46. 

(2,  -  h). 

58. 

b{b  +  4a^)  =  0. 

48. 

(0,  -  8). 

59. 

62(27a<  -6)  =  0. 

49. 

(1,  -  27). 

60. 

¥  -  27  a*  =  0. 

51. 

0.45,  1.80,  -  1.25. 

61. 

See  Ex.  23,  Chap.  I. 

CHAPTER  VII 


Page  155 

5.  x2  +  2/2  ±  2ax  =  0. 

6.  x2  +  2/2  -t  2  ax  ±  2  ay  +  a2  =  0. 

7.  x2  +  2/2  +  3  x^  2  2/  =  0. 

8.  (-  2,  5);  V6o. 

9.  (-2,  3);  2  V3. 
10.   (3.,  -  1) ;  1  Vlil. 

11-   {~hi)>  0. 

14.  x2  +  2/2-3x-32/  =  0. 

15.  X2  +  2/2-  3x  -4  =  0. 


16.  x2  +  2/2  +  26  X  +  16  2/  -  32  =  0. 

17.  x2  +  2/^  -  5x  +  4  y  -  46  =  0. 

18.  x2  +  2/2- 20x- 202/ +  100  =  0, 
x2  +  2/2-4x-42/  +  4  =  0. 

19.  x2  +  2/2  -  12x  -  12  2/  +  30  =  0, 
25x2+25  2/2 -l-60x-602/  +  3G  =  0. 

20.  x2  +  2/2  +  2x  + IO2/+ 1  =  0, 
a;2  +  y2  _  12  X  -  4  2/  +  15  =  0. 

21.  2x2  +  22/2 +  6x  + 3  2/ -10=0. 


Page  156 


22. 

x2  +  2/2  +  22  X  -  34  2/  +  121  =  0, 

30. 

x2  +  2/2-2x-  102/  +  1  =  0. 

31. 

23. 

X2  +  2/2-  10x-28  7/  +  217  =  0. 

32. 

24. 

x2  +  2/2  +  22  X  -  44  y  -  20  =  0, 

33. 

x2  +  2/2  +  2  X  -  4  2/  -  20  =  0. 

34. 

25. 

4x2  +  42/2  ±7y- 36  =  0. 

35. 

26. 

7x2+  I6y2_ii2  =  0. 

36. 

27. 

9x2 +  5y2_  45  =  0. 

37. 

28. 

5x2  +  92/2-180  =  0. 

38. 

29. 

I  V385,  i  V105. 

39. 

40.  ^  V2,  i  V3;  (± 

iAo) 

age 

157 

41. 

5x2-42/2-20  =  0. 

48. 

42.  3  2/2 -x2-  12  =  0. 

43.  28x2-36  2/2-  175  =  0. 

44.  24x2-252/2-384  =  0. 

46.  3x2  -  2/2  -  3a2  =  0. 

47.  8x2-2/2-16  =  0, 
8y2_a;2_  124  =  0. 


16  x2  +  25  2/2  -  400  =  0. 
5x2  4.9y2_80  =  0. 
16  x2  +  25^2  _  400  =  0. 
196x2  +  132  2/2  -  14553  =  0. 
4,3;  J  V7;  (±V7,  0). 
J;  3x2  +  4^2  _  3^2=  0. 

6x2  +  9^2  _  405  =  0. 
a;2  +  4  2/2  _  a2  ^  0  ;  },  V3. 
\  V2. 

3x2  +  52/2-30  =  0. 
;  2  X  ±  Ve  =  0. 


48.  x2  -  2/2  -  21  =  0. 

49.  x2  -82/2  +  4  =  0. 

50.  25x2  _  144  2/2  _  3500  =  0. 

51.  252/2-  9x2-  16  =  0 

52.  cos- 


53.  ^  V5,  i  V5. 


ANSWERS 


360 


54. 


58. 


Page 
61. 
62. 
63. 

71. 
72. 

Page 
80. 
81. 
82. 

Page 
94. 
95. 


i -v^;  (±v^,  0); 

2x±5y  =  0. 

iVl3;  (±VT3,  0);  13x±9  Vl3 

158 


55.  52x2  _ll7y2_  576  =  0. 
57.  3x2-41/2-84  =  0. 
0;  2x±3y  =  0. 


2^?  V2. 


67.  y  ±  5x  =  0. 

68.  x2  _  82/2  _6y  + 9  =  0. 

69.  2/2  +  4?/-2x  +  11  =  0. 


64.  x2  +  y2  _  5px  =  0. 

65.  x2  +  2/2  +  3x-6y  =  0. 

66.  7x-  32/  +  2  =  0. 
70.  91x2  +  842/2  -  24X2/  -  364x  -  lb2y  +  464  =  0 

2/2 -10x  + 25=0.  73.  4x  +  32/-31=0,  4x+ 32/  +  19  =  0 

(2/  -  2)2  ±  x8  +  x2  =  0.  74.  5  X  +  y  -  5  =  0,  X  -  5  2/  +  7  =  0. 

159 

Circle.  83.  Concentric  circle. 

Circle.  84.  Straight  line. 

Circle.  85.  Straight  line. 

91.  Parabola. 


86.  Circle. 

87.  Two  straight  lines. 
90.  Parabola. 

93.  Two  parabolas. 


160 

Circle. 
Parabola. 


96.  Hyperbola. 

97.  Parabola. 


98.  Hyperbola, 

99.  Witch. 


101.  8i)V3. 


CHAPTER  VIII 


Page  175 

1.  (1,  1),  (-  2,  3). 

2.  (0,  1). 

4.  (0,0),  (-1,  -2). 

5.  (1,  21). 

6.  (1,3),  (I,  -1). 

8.  (2  ±  VB,  2  1^  V6). 

9.  ^6,  Vl3. 

Page  176 

18.  (2,  2). 

19.  (0,0),  (±V2,  ±i>^) 

20.  (2,  2),  (f,  -  I). 

21.  (0,  0),  1^- 

22.  (2,  1). 


.  +  »l2       1  +  ,„2 


10.  2x-?/  +  2  =  0. 

11.  -3. 

12.  2x  +  32/±6V2  =  0._ 

13.  bx-ay  +  ab±  ab  V2  =  0. 
16.   (0,  0),  (4,  li). 

16.  (±lf,  ±5),  (±li,  ±3J). 

17.  (I,  0),  (1,  -  1). 


23.  (0,  0),  (-1,  0). 

24.  (1,  ±2V3),  (6,  ±6V2). 

25.  (±  I-  V48  T  3  Vl3, Q )• 

26.  {b  -  a,  ±  2  Va6). 

27.  (±2a,  a). 


28. 


29. 
30. 
31. 

Page 
35. 
36. 


(0,0),  (fa,  ±faV2). 
(2  a,  a). 
(±2a,  a). 

177 

3x  +  y  -I  =  0. 

5  a;2  +  5  2/2  +  28  X  +  42  2/ 


(0,  0),  (-  2  a  +  2  a  A  ±  2  a  V2  Vi  -  2) 


32.  (±2 a,  a). 

33.  5x  +  42/  +  6  =  0. 

34.  2x  +  5y-13  =  0. 

38.  3x2 +3^2  + 13a; +13?/ -4  =  0. 

39.  x2  + 82/2 -9  =  0. 


370 


ANSWERS 


CHAPTER  IX 


Page  209 

1.  9x2  +  14x  +  6. 

2.  20(x  +  l)(3x2  +  6x  +  2). 
2a 


(X  +  o)2 
6x2 


(X8  -  1)2 

14  (1  -  X) 
(3x2-6x  +  l)a' 
1 


^^^^V|-^> 


(X  +  1)2 

3Vxi 
8.  2(4x 

9 


3)  +  _(6_7x). 


x  +  1 


2xVx 

10.  l/A_JL_J_  +  _^> 

A^x      <^2      a;^x      x^x2y 

11.  2(3x2_5x+6)(6x-  5). 

12.  6x(x2  +  l)2. 

IS  «*  +  ^ 


14. 


2  V4x2  +  6x-6 
2x  +  1 


16.  - 

16.  - 

17.  - 


3  v^(x2  +  X  -  1)» 
2x 


(X2  +  1)2 

6(3x2  ^-2x) 

(X8  +  X2  +  1)2 

5x 


\^(X2  +  1)6 

18.  5(2x-l)(2x-3)(x  +  l)2. 

19.  (3x- 6)  (12x2 -55  X +31). 
20    2^'  +  ^  +  l 


21. 


Vx2  +  1 

3x*-10x3  +  6x2  +  x-2 


(x2-4x  +  3)*(x3  +  l)l 

22.  V_L=-f-i=V 
2VVx  +  l       Vx-1/ 

23.  1  +        ^        . 

Vx2  +  1 


24.  2x/ 


25. 


1 


1 


30. 


31. 


32. 


33. 


34. 


41. 


42. 


.\^(3X2+1)2         \/ (3x2+1) 
1 


.) 


(X  +  l)Vx2-l 

X  +  1 


27. 


28. 


29. 


(X  -  1)2  Vx2  +  1 
X-X2 
(X2  +  l)i(x8  +  1)* 

2x2  +  1 


Vl  +  X2 


-2x. 


a-' 


(a2  -  x2)^ 
X  —  Va2  +  x2 


cfi  Va2  +  x2 
4  X  (2  y2  _  x2) 
2/(3  2/ -8x2)' 
5x^-3x2 

52/* -1 
3  [x2y*  +  (X  -  2/)2] 
3  (X  -  2/)2  -  4  x^ys  ' 
3x2- 5 x< 

4y8_i 


35. 


36. 


37. 


38. 


X  +  Vx2  -  2/2 


y 


V 


X  -  2/(x  -  2/)2 
_6x.        25 
2  y '        2  2/3 
x6         6  a^x^ 


y 


18 


6* 


39. ;   - 

40 


6a*y 


3  2/2  -  a2     (a2  -  3  y^f 
3x2        6x(l-32/2) 
32/2 +  r     (1  +  32/2)3   ' 
2(3x2-2  2/).        128x2/ 


3  j/2  +  4  X       (3  2/^  +  4  x)3 


ANSWERS 


371 


Page  210 

43.  tan.,  3x  +  y  +  6  =  0; 
nor.  ,x  —  3y-\-b  =  Q: 
tan. ,  3x  —  2/4-9  =  0; 
nor.  ,x  +  3y  —  7  =  0. 

44.  X  -  6  y  + 17  =  0,  6  X  +  2/  -  9  =  0. 

45.  x  +  2y  -  2  =  0. 

46.  4x-3?/-l  =  0. 

47.  x  +  2y  -1  =  0. 

48.  (Sy^-xi)y-yiX-Xiyi-Sa=0. 

49.  X  -  3  y^y  +  2  Xi  -  3  =  0. 

50.  2yiy  -3x^x  +  xl  =  0. 

51.  (2  -  xf)x  +  xfy  -  3xi  =  0. 

52.  xi~*x  +  yi~^?/ =  a*. 

53.  Xi~^x  +  2/i~i?/  =  a^. 


54.   52/  +  Vl0x-6  VH  =  0, 

lOy-  5V1OX  +  4  V5  =  0. 

tan.,  4x  —  2/  —  6  =  0; 

nor.,  x  +  4y  —  10  =  0: 

tan.,  4x  —  2/  +  6  =  0; 

nor. ,  X  +  4  2/  +  10  =  0. 

X 


65 


66 


dy 
ax 


61, 


dy 
dx 
(-0.57,2.08) 


Page  211 


62. 

(±ia  V2,  ± 
(1        ''       , 

lbV2] 

76. 

62          N^ 

a22/ix  -  62xi2/  =  0 ; 
o262 

63. 

^6^x2 

TT       . 

-,  tan 
4 

73. 

V      Va2  +  62 

Vp(p  +  Xi). 

2+62/ 

77. 

+  a*y{ 

age 

212 

78. 
79. 

80. 

0,  tan-i  |. 

TT 
2' 
TT 
3' 

86.  -,  tan-i|. 
2              * 

86.  tan-13. 

88.  -,  tan-i  V2. 
2 

90.  tan-i|. 

91.  0,  tan-isVs. 

92.  0, -,  tan-ijv^. 

93.  The  length  is  twice 

81. 

tan-1 5  Vs. 

89.  tan-1/5. 

the  breadth. 
94.  He  walks  2.86  mi. 

Page  213 

95.  8rd.,  12  rd. 

96.  Cross  section  is  a  square. 

97.  Of  equal  length. 

98.  4  mi.   from  nearest  point  on 

bank  to  A. 

99.  jFD  =  (V2  -\)AB. 


103.  Area  of  ellipse  is  -  area  of 

rectangle. 

104.  Central    angle    of    sector    is 

|7r  VO. 

105.  Breadth  a,  depth  a  Vs. 

106.  Breadth,  f  a  V3;  depth,  fa  V6. 


Page  214 

107.  Velocity  in  still  water  \a  mi. 

per  hour. 

108.  Radius   of    base   equals   alti- 

tude. 

109.  Altitude  is  |  radius  of  sphere. 


110.  a- 


6m 


Vn2  -  m2 
6n 


mi.  on  land, 


Vna" 


mi.  in  water. 


372 


ANSWERS 


111.  Altitude    is    J  '^    radius    of 

semicircle. 

112.  Altitude  is  jr  distance  between 

vertex  of  pai-abola  and  bound- 
ing straight  line. 

113.  (&,  a). 

114.  (±|V3,  I). 

115.  maximum  value  when  x  =  i, 

minimum  value  when  x  —  \; 

points  of  inflection  when  x=  —  1 

1±V6 

or 

5 

122.  Stationary  when  <  =  0,  4,  or 

moving  backward  when  4  < 

123.  20,  10  V5;  (100,20). 


116.  minimum  ordinate,  x  =  — -\ 

V3 

maximum  oixlinate,  x— — 


points    of    inflection 
(±  a,  0). 
117.  (±i  V3,  I). 

/±  -  V27  -3  V33, 
\      6 


V3 
(0,   0), 


119 


±  :^  V5  V33 
12 


120.(±|V^,±^>^). 

121.  (1,  3),  (5,  -  5). 
8  ;  maximum  velocity  when  t 
t<8. 


rii) 


=  1. 


Page  215 

124.  Whenx=  i;  parallel. 

126.  Velocity  of  top:  velocity  of  bottom  =  distance  of  bottom  from  wall 
distance  of  top  from  ground, 

126.  54 TT  cu.  ft.  per  hour.  132.  3 y  =  x»  +  9z  -  19. 

127.  389  ft.  per  minute.  133.  Sx-2xy  -2  =  0. 

128.  41.9  ft.  per  second.  134.  3y  =  2x^  +  U. 


129.  15  ft.  per  second. 

130.  0.2  hi.  per  .second. 

131.  17.9  mi.  per  hour. 


136.  -  -1  =  A;(l-x). 

y 

137.  90,000,  677iV- 


Page  216 
138.   108. 


140. 


hk 
n  +  1 


141.  J  «2.  142.  851.  143.  K^  a^.  144.  10|.  145.  17  1 


CHAPTER  X. 
Page  225 

1.  {-  1,  5),  (-  7,  7),  (2,  -  5).  2.  a;2  +  4  2/2  -  4  =  0. 

3.  2/^  -  15  ?/2  -f  3  x2  +  75  y  -  6  X  -  106  =  0. 


4.  62a;2  ^  a2y2  _  2  a^by  =  0. 

5.  62x2  4.  a^yi  _  2  ab^-x  =  0. 

6.  62a;2  _  a2y2  -  2  ab^  =  0. 
X  (a  +  x)2 


7.  2/2  =  - 
Page  226 


2a  +  X 


10.  y  = 


4  a3  -  ox" 

x2  +  4  a2  ' 


11. 


y.^      (2a  +  x)8 


8. 


2/''  = 


9.  2/  = 


(X  -  a)2(2  a-x) 


2ax2 


xa  +  4  o2 


12.  2/^  =  (^±^. 

a-x 

13.  y^  =  4px  -f  4p* 

14.  2/2  =  4px  —  4j)'. 


ANSWEES  373 

16.  2x2  +  32/2-6  =  0.  ■  4a      4  6 

n.ah-c.  20.  x2  +  9^2  + 6a;_36y +  36  =  0. 

21.  196  x2  +  900  2/2  +  784  X  +  5400  y  +  8875  =  0. 

22.  x2  -  42/2  _  2x- 162/- 19  =  0.  25.  x2  -  8x  +  I62/ -  64  =  0. 

23.  2x2-62/2-8x  +  36y-47=0.  26.  3x2  +  42/2- 12x  -  242/ -  27=0. 

24.  2/2  +  42/-8x  +  28  =  0.  27.  8x2  +  92/2  -  16x  -  64  =  0. 

Page  227 

28.  5x2  -  42/2 +  10x- 162/ -31  =  0. 

29.  2/2  +  4  2/  -  X  =  0. 

30.  i  V5j  (-  1,  2) ;  (2,  2),  (-  4,  2) ;  (-  1  ±  V5,  2) ;  6x  +  5  ±  9  V5_=  0.    " 

31.  I  VlO;  (-3,  2);  (-3±V5,  2);  (-3±V2,  2)  ;  2x  +  6  ±  5 -^2  =_0. 

32.  i  Vl3;  (3,  -4);  (5,  -4),  (1,  -4);  (3±vT3,-4);  13x-39±4  Vl3  =  r,  ; 

3x-2y-  17  =  0,  3x +  22/-  1  =  0. 

33.  1  VlO;  (-1,  2)_;  (-1  ±V2,  2);  (-  1  +V5,  2);  5x  +  5i:2  V5  =  0; 
V3(x  +  l)±V2(2/-2)  =  0. 

34.  (-1,  ^);  (-^,  -I);  3x  +  l=0;  82/-7=0. 

35.  {-  2,  -  3)  ;  (-  I,  -  3) ;  2/  +  3  =  0  ;  4x  +  13  =  0. 

36.  (i  V3,  i),  (i,  -  1  V3),  40.  x2  +  14  2/2  -  14  =  0. 

/I  +  V3     1  -  V3\  *2.  xy  =  -  18,  or  xy  =  18. 

y ^'   — 2 — ;■  ^^-  17x2  +  7  2/2-2  =  0, 

37.  x2- 42/2 -4=0.  or  7x2+  172/2-2  =  0. 
,„      ,      x2(3aV2-2x)                             44.  x2±2/  =  0,  or2/2±x  =  0. 
^^.y^  =  -^ ^ : 45.2x2-2/2-1  =  0. 


3  a  V2  +  6x 


Page  228 


46.5x2  +  8  2/2  =  40.  48.   1^  +  VV  ^(^^^  -  ^  +  V 

47.  4  X2/  =  7.  ■  \x  -  2/7       aV^  +  x-y 

49.  5x2-62/2  =  30. 


CHAPTER  XI 
Page  244 

1.  Hyperbola;  center,  (—7,  2);  slopes  of  axes,  2  and  —  \. 

2.  Parabola;  slope  of  axis,  \;  vertex,  (||,  —  4if). 

3.  No  curve. 

4.  Hyperbola ;  center,  (—1,0);  slopes  of  axes,  1  and  —  1. 

5.  Hyperbola;  center,  (2,  —  |);  slopes  of  axes,  1  and  —  1, 

6.  The  line  x  —  2/  +  1  =  0  taken  twice. 

7.  Elliijse;  center,  (—  1,  2);  slopes  of  axes,  1  and  —  1. 

8.  A  pair  of  straight  lines  intersecting  at  (3,  —  2),  and  having  the  slopes 

- 1  ±  i  Ve. 

9.  A  pair  of  straight  lines  intersecting  at  (3,  —  2),  and  liaving  tlie  slopes  § 

and  —  |. 
10.  Parabola;  slope  of  axis,  1;  vertex,  (—  4^,  —  \\). 


374  ANSWERS 

11.  The  parallel  straight  lines  x  +  3y-5  =  0,  x  +  3y-l  =  0. 

12.  Ellipse;  center,  (2,  -  1);  slopes  of  axes,  f  and  -  |. 

13.  Point,  (0,  2). 

Page  245 


20.  tan-i : 

A  +  B 

23.  2x2  +  3xy  +  2/2  +  12x-13?/-50  =  0. 

24.  xy  -  2r/2  _  2x  +  4y  =  0.  25.  x^  -  xy  +  y"^  -  a"^  =  0. 

26.  6x2  +  5x2/  +  2/*  -  29x  -  13?/  +  30  =  0. 

27.  9x2-12xy  +  4y2_ii7x  +  78y  +  380  =  0, 

or  49x2  _  66 xy  +  16i/2  -621x  +  354y  +  1964  =  0. 

V. -Jtan2^-1 
1  -  tan2^  if  tan  ^  <  1, if  tan  ^  >  1  (/3  the  angle  between 
2             2                  ^  2 


28 


tan- 
the  lines).  2 


CHAPTER  XII 
Page  262 

1.  6x-y  +  3  =  0;  (0,3),  (-1,-2).  5.(0,3). 

2.  X  +  y  -  3  =  0  ;  (0,  3),  (1,  2).  6.  (|,  -  \). 

3.  X- 2/ +  1  =  0;  (-1,1).  7.(2,3). 

4.  2/-2x  +  6  =  0;(l,-3),(2,-l).  8.  (1,  -2). 

Page  263 

9.  8x-2y  =  0,x-2/  +  l=0.  14.  x  -  3?/- 2  =  0,  2x  -  2/  +  l  =  0. 

10.  x  =  0,  x-2/  +  l=0.  21    -. 

11.  3x  +  2/-l  =  0.  ■  a2 

12.  2  X  -  2/  =  0,  X  +  2  2/  -  10  =  0.  24.  At  infinity. 

18.  X  +  2  2/  -  2  =  0,  X  -  3  y  -  2  =  0.  25.  6ex  -  oj/  =  0,  bx  +  aey  =  0. 

Page  264 

30.   -e.  81-  0^^  =  2  62. 

Page  265 

47.   (ae,  ±  ^)  ;  tan-J(±e). 

CHAPTER  XIII 
Page  297 

21.  a  cos  2  ax. 

22.  o  [sec*  (ax  +  h)  ctn  (ax  +  c)  -  tan  (ax  +  &)  esc*  (ox  +  c)]. 
28.    -8csc24x.                                 26.  sec2x. 

2(2ctn2x  — 1)                           27.  mn  sec™ nx  esc"  wx (tan  ?ix  —  ctn  mx). 

csc2x                                  28.  2sec2  2x(2tan2x  +  l). 

3sec3x(tan3x-l)                  29.  -  2  csc2x(2  csc24x  +  ctn  4xctn  2x). 

(tan3x  +  l)2                        30.  cos(xcosx) (cosx  —  xsinx). 


ANSWERS  375 


31.  5sin2xcos3x.  49.  2(x  +  l)e^  +  2»:. 

32.  8  sec^x  —  3  sec  x.  2  a^ 

33.  — —  cosx. 
V2 


X*  —  a* 


51.  e^T^^^cF^^i-^^^ ^—\ 

V(l  -  X2)*         Vl  -  xV 


34.  — ==  sin  Vl-x2.  V(l  -  x2)^ 

36.  secxtanx.  Vx^Tx 

^■^,-  .,...VH(i-!5?). 

37.  .  64.  a'ana^loga- sec^x. 

Vl-x2  55.  2x{l  +  21ogx). 

gg 1  66.  loga-sec2x(sec2x  +  2tan2x)at''n3-sec»a:. 

(1  +  x)  Vx  67.  2sec2x(sec2x  +  loga- tan22x)a«e«''^ 

1  68.   [2{a  +  x)sinmx  +  m  cosmx]e(«  +  =')^ 
69.   -2. 


39 

Va2  -  x2 

40. 


a  +  X  \x  ^      ^  ' 


-,  1  ox.   

41.    — •  e2a:  +  e-2x 

(x  +  1)  vx 


1  61. 

62. 
63. 


42. -•  3  +  5sm2x 

(x2  +  2  X)  Vx2  +  2  X  -  1         ^^  2 

43.  0. 


44.  ^^ 64.  e^<:os«cos(a  +  xsina). 

65. 


45.  -•  sm(2x2  +  2a2) 

66.  cos-^x. 


46. 
47. 
48.    - 


a  +  b  cos  X 
2 

Vl-X2 
1 

(2x-l)  Vx2-x 

1 

2x2  +  2x  +  l 
4x 

CO 

4x 


67.  sec  ax. 

68.  (a2  +  l)e««i""*'". 

69.  ctn-ix.  . 

cos-^x 
70. 


V(l  -  X2)8 

X*  +  1  71.  ef^sinmx. 


Page  298 

72.  2xctn-i«.  ^g_(x-l)log(x_D. 


*  (x2-2x)3 

sin-ix  1 

— :;i 77. 


78. 

a  +  b  cos  X 
74.  ^         _         1  78.  4csc(4x  +  2)[l-ctn(4x  +  2)]. 

Vx2  -  Ofi         Va2  -  X2  Yg ^  . 


76 


1     jx  +  a 
X  \x  —  a' 


V2  ax  -  a;2 
80.  2  Va2  +  x2. 


376  ANSWERS 

sec- 12  Vx  91.  2/x^(l  +  loga;). 

•   Vx(4x-1)3'  92    yrtan-i(a  +  x)  ^    log(a  +  x)-| 

82-   V2ax-x2.  L       «  +  ^            l+(a  +  x)d' 


83. 


2  -  Vx2-  1  2/  (tan  xy ) 


2x(x2-l)  ■  93 


84.    - 


,  log  X  —  X  tan  xy 


y 


(e^  +  2)  Ve2^+  2e^-l  ^^^fTxa 

(e2x+e-)log(e^  +  2)^  94.  ^^^^^^_,^ 


V(e2a:  +  2ex_i)3  ^^    y  C08X  +  sin (x  -  y) 

85.  X  tan-i  Vl  —  x^.  '     sin  {x  —  y)  —  sinx 

86.  ^^i±^'.  '  96.         '"^ 


2  +  x2  {ny  +  \)x 

V  r  r  nH    e2' sin  X  +  e^  sin  w 

87.  -^(sec2  Vxlogsin  Vx  +  l).  97.    ^. 

2  Vx  ^  °  01  cos  X  —  e*  cos  y 

y  a»    y(l -cos x)- cosy 

88.  i;^(xctnx-logsinx).  M-  ^i^^  _  ^(i  +  ,5^^)'  . 

89.  yx^fl  +  logx  +  (logx)2l.  99.  ^^^^V  -V 

L*  J  X  log  X  —  x2  cos  y 

90.  ye^(l  +  logxV  IQQ.  Vd-^^-y^) 

\a^  /  x(l+x2  +  y2) 

Page  299 

101    ^  +  ^,  2(x2+y2)     4(x  +  2y)(x2  +  y2) 

■  X  -  y      (X  -  y)3  (x  -  y)5 

102.  -  e^-y,  -  e*-J'(H- e^-p),  -  e^-2'(l  +  e^-»)  (1  +  2e*-!'). 

103  ^-y'    2(x  +  y)(y2-l)     6 (x  +  y)2(l  -  y^) 

■  X2  -  1  '  (X2  -  1)2  '  (pfl  -  1)3 

104  ^  +  y -^        4 (X  +  2/)       8(x  +  y)(l-2x-2y) 

■  x  +  y  +  l'  (x  +  y  +  l)3'  (x  +  y  +  l)6 

jQg         y-g         y(l+logx)-2x    2y[l  +  logx  +  (logx)2]-3x(l+logg) 

■  x(l-logx)'     [x(l-logx)]2   '  [x(l-logx)]3 
-  1  ±  V33 


106.  X 


8 


107.  1  or  2.  ,,„  ,  o ;      ,  '^ 

112.  X  =  «7r,  X  =  2  KTT  ±  -  • 

108.  tan-i2V^.  3 

113.  (ae^  fae-^). 


109.  tan-i(2tan-sec-V 

\  2       2/ 

110.  tan-4,  tan-13. 


114. 


(^^--*) 


116.  Maxima  when  x  =  (2  A;  +  1)-,  minima  when  x  —  kir]  points  of  inflection 
-when  X  =  (2fc  +  !)-• 

117.  Maxima  when  x  =(2fc  +  |)7r,  minima  when  x  =(2*;  +  |)7r;  points  of  in- 
flection when  X  =  kit. 


ANSWERS 


377 


118.  Maximum  when  x_=  n,  minimum  wlien  x  =  0,  (n  =  2fc):    points  of 
inflection  when  x  =  n  ±  Vn,  (n  ?i  1) ;  x  =  2,  (n  =  1);  x  =  0,  (n  =  2 k  +  1). 


119.  Circle. 


120.  2  V(s  -  3)  (5  -  s),  4(4-8). 


Page  300 
122.  a. 
123. 


ab 


128. 


500  sin  a 


; ,  where  x  is 


2  V62_a2t2• 


126.  -bsind 


62  sin  5  cos  ^ 


times 


Vl0,000-x2sin2a 
the  distance  from  the  center. 
1 29 .   1 5  sq.  ft. ;  9. 03  sq.  ft.  per  second. 


Va2- 

62  sin2  e 

130. each. 

velocity  of  AB. 

1  where 

2       2 

e  =  CAB. 

131.  23.7. 

500 

132.  At  an  angle  tan-^fc  with  the 

127. 

,  wl 

lere  x  is  the 

Vl0,000-x2 

ground. 

distance  from  the  center. 

Page  301 

- 

133. 
134. 

Sin. 
1:V2. 

145.  log-  =  ft(x-a). 
6 

135. 

5V5ft. 

146.  s  =  ce^'. 

136. 

2. 

147.  A7r  +  (-l)«^.6661. 

137. 

V2-I. 

148.  2fc7r  ±  .567,  2kTr  ±  2.206. 

138. 

2. 

149.  ktr. 

139. 

e-1. 

150.  fcTT,  (2A:  +  l)j,  2kir±1- 
4                  0 

140. 

e 

151.  A;7r  ±^,  ^^'ri  -• 
4               0 

141. 

log  2. 

142. 

X  =  2  a. 

152.  4.4934. 

a  j  '^         *\ 

163.  4.275. 

143. 

-  (ga  _  e  «)  -f  c. 

154.  0.199. 

« 

155.   -0.7085. 

144. 

2/  -  3  =  &  log  - . 
2 

166.  1.857,  4.54. 

CHAPTER  XIV 


Page  323 


2.  tan.,  X  -  iy  +  p«2  =  0  ;  nor.,  tx  +  y  -2pt-  pt^  =  0. 

4p  4p      " 

3.  X  =  — ^»  2/  = 

<2  '  ^         i 

4.  tan.,  <2x  _  2  «y  +  4p  =  0 ;  nor.,  2  t^x  +  t^y  -  ipt^  -Sp  =  0. 

ab  ,         aht 

6.  r.=z±  ——  y  =  J:— — 

y6-2  +  a2«2  V62  +  a2«2 

_  a(62-  OT2a2)        _     2a62m    ^ 
■  ^  -     62  +  m2a2    '  ^  ~  62  +  m2a2 ' 

.  „  2asin3(^ 

7.  x  =  2asm2<^,  y  = 

COS0 

10.  (l  +  3«2)x-2i3y-2a=:0. 


378  ANSWERS 

Page  324 

■  ^  2x-a  3i2  +  l 


12.  a;3  +  1/3  _  3  oxy  =  0.  16.   (3  y  -  x)2  =  2  Vx2  -  2/2. 

13.  2/2  (ax  -  a2)  =  x2  (a2  +  A;2  -  ax).  17.  9  y  =  (x  -  ?/)2  -  6  (x  -  2/). 

14.  3  2/2  -  4  xy  +  2  2/  -  1  =  0.  18.   (x2  +  y^)  (ox  +  62^)  =  cxy. 


/—    -  —  tan' 


■ll 


19.  Vx2  +  2/2  =  a  V2  e^ 

o^  a/.„^^\.  2x  +  a     ja-x 

20.  X  =  a  cos2  - ,  2/  =  -  I  sin  ^  +  tan  -    ;   ?/  = \  ~ 

22\  2/  2\x 

21.  x  =  (a  —  c  tan^)  sin2^,  2/  =  (acosfl  —  csin^)sin^;  2/(x2  +  2/2)  =  x(a2/ -  ex). 

22.  X  =  -  (a2  +  A;2  cos2  ^),  y  =  -  {a^  tan  »  +  fc2  gin  0  cos  fl)  j 
a(x-  a)  (x2  +  2/2)  =  i2a;2. 

a{a2  —  x2) 

23.  X  —  a  tan  ^,  2/  =  a  cos  2  ^ ;  2/  =  -^; r-^  • 

a2  +  x2 

Page  325 

24.  X  —  a  sin  d  (cos  ^  +  sec  6),  y  =  a  cos  ^  (cos  d  +  sec  tf) ; 
2/(x2  +  2/2)  =  a(x2  +  2  2/2). 

29.  6x  cos  0  +  a2/  sin  <^  —  a6  =  0. 

30.  !^.  31.  !r  +  ^.  32.  ^sin-i?-^ 
4                             4      2  2  «„^ 


8,.  ,,„.,'l±:^<±lMs:^\ 

gb 

34.  X  =  a  cos  ^  +  i  V62  _  a2  sin"^  S,    y  =  {I  —  I)  a  sin  ^,   where  the  center  of 

the  driving  wheel  is  the  origin,  a  the  length  of  the  radius  of  the  driv- 
ing wheel,  b  the  length  of  the  connecting  rod,  and  lb  the  distance  along 
the  rod  from  the  wheel  to  the  point. 

35.  Straight  line. 

37.  2  aw  sin  -  ;   w  Va^  —  2  aAcos  0  +  ^2^    where   u  is  the  constant  angular 


velocity. 


Page  326 


38.  2  au  sin  - ,  w  v  a'*'  —  2  ah  cos  6  +  h^.    ■ 

2 

40.  adw,  where  a  is  the  radius  of  the  circle,  ad  the  distance  through  which 

the  point  of  the  string  in  contact  with  the  wheel  has  moved  along  the 
rim  of  the  wheel,  and  w  the  constant  angle  of  velocity. 

41.  X  =  a(cos2e  +  sin2tf),  2/ =  a(l  +  sin2^  -  cos2d) : 

2  V2  aw,  2  a  (cos  2  ^  —  sin  2  ^)  w,  2  a  (sin  2  ^  +  cos  2  ^)  u. 
^„      „      x2(3a  +  x)  48.  The  witch. 

43.    1/2  =:  ^ • 

a  —  x  49.  X  =  a  (cos  6  +  0sm  0), 
43.  Circle.  y  =  a  (sin  0  —  0  cos  0). 

46.  ?/ (x2  +  2/2)  =  2  a  (x2  +  2 2/2).  50.  x  =  a(l  4- m2),  ?/ =  ma  (1  + ?n2)  ; 

47.  y  =  x  —  a.  ay^  =  x2(x  —  a). 


ANSWERS 


379 


Page  327 

53.  Ellipse. 

54.  Hyperbola. 

55.  Straight  line. 

56.  Concentric  circle. 


57.  z  +  2p-0,  py^  =  afi. 
59.  Concentric  ellipse. 
61.  Ellipse. 


Page  328 

65.  Parabola. 

67.  Concentric  circle. 

70.  Concentric  circle. 

TTX 


71.  y  =  xctn 


2a 


72.  x  =  a{4)  +  sm<t>), 
y  =  a{-l  +  cos^). 

74.  8  a. 

8a{a  +  b) 
b 


75 


Page 
26. 

27. 

28. 


349 


CHAPTER  XV 


(l  0353  a,  ^y 

(l^-l)'(i^'T)- 


29.  (0,0),(a,±^),(a,±^) 


80.  Circle. 
31.  r  =  acos2^. 
a  cos  2  0 


32.  r 


coa^9 


Page  350 

40.  r^  sin2(?  cos  ^  +  (2  a  +  r  cos  6)^  =  0. 

41.  r  cos 61  =  a  cos2  (?,  (r^  +  a^)  cos^  +  2  ar  =  0 
49.  (x2  +  2/2)3  _  4  a-^x^y^  =  0. 

60.  (x2  +  y^  +  ax)2  -  a2(x2  +  y"-)  =  0. 

61.  (a;2  4-  2/2)2  _  2  a2  (x2  _  2/2)  +  a*  -  6*  =  0. 

82.  {X2  +  2/2  _  ax)2  =  62(a;2  +  2/2). 

63.  (X  -  a)2  (x2  +  2/2)  =  &2x2. 
54. 


log(x2  +  y2)-2atan-i^- 


56.  25,000,000. 

57.  1,200,000,  or  4,800,000. 


Page 
59. 

61. 
62. 
63. 


2c 


64.  r  = 


351 

r  =  - 

1  —  costf 

Straight  line. 

Circle. 

Circle. 

2  ce  cos  ^ 


65.  r  = 


66.  2r  = 


ce2  cos  d 
—  eP'CO&^B 

ce 
1  —  e  cos  B 


72.  «/'(«),    «  Vt/(«)]2  +  [/'(fl)]2. 


1  -  e2  cos2  ^ 
78 


^r(e) 


V[/(e)]2  +  [ne)f    V[/(fl)]2  +  [/'(<?)]« 


380  ANSWERS 

Page  352 

74.  w  Va2  +  62  ^.  2  ab  cos 6.  ^9    J_(e2.a_i\ 

75.  Spiral  of  Archimedes.  4  a 

76.  2a2.  80.  \ira^. 

tt.  ia2.  81.   ia2  (4 -■»■). 

78.   i7r%2.  82.   i,r(a2  +  2  62). 

83.  8  a. 


CHAPTER  XVI 
Page  362 


a 


2    ox'  (8  g  -  3  x)^ 


4.  \  a,  a. 
a? 
3(2a-x)2    ■  *•  36" 

Page  363 

6.  fa2,  |a2.  ,o    8   ■    « 

-X,  13.  -sm 

7.  e  '^.  3       2 

8.  0,  \.  ,4    „,__4 


10. 


14.  y2_ (a;  _  2r>)s, 

a(6-4co8g)8  27p^ 

9-6cos«  i«    a  Va2  -  .r^ 


3  fl  ^ 

11.  -asin2-;  greatest  value,  -  a  ; 


15. 


-U.0111--;  t-iettLest  value, -a  ,o     ,  .j  ,i       .-i 

4  3^  '4     '  18.  (a;  +  2/)J  +  (j--  ?/)1^  2.0 

least  value,  0.  21.  «2  ^  ^2  _  ^  _  q. 


12.  2-^. 
3r 


Page  364 


23.  Minimum  curvature  when  x  = 

24.  Maximum  curvature  when  x  =  (2  A;  +  1)  —  ; 

minimum  curvature  when  x  =  krr. 

25.  Maximum  curvature  at  ends  of  major  axis; 

minimum  curvature  at  ends  of  minor  axis. 

L        \dx/  J  dx3         dx  \dxV 
28.  (2fe  +  l)|. 


INDEX 


[The  numbers  refer  to  the  pages.] 


Abscissa,  36 

Acceleration,  202 

Addition  of  segments  of  a  straight  line, 

32 
Algebraic  functions,  43,  121 

differentiation  of,  178 

implicit,  188 
Angle  between  two  lines,  57 

between  two  curves,  211 

eccentric,  304 

vectorial,  329 
Angles,  65 
Arc 

length  of,  195 

limit  of  ratio  to  chord,  195 

derivatives  with  respect  to,  197, 347 
Archimedes,  spiral  of,  332 
Area,  204 

of  an  ellipse,  304 

in  polar  coordinates,  348 
Asymptote,  128 

of  an  hyperbola,  145 
Auxiliary  circle  of  ellipse,  304 
Axes 

of  an  ellipse,  141 

of  an  hyperbola,  145 
Axis,  of  symmetry,  121 

of  a  parabola,  147 

radical,  175 

Bisection  of  a  line,  39 

Cardioid,  337 
Cassini,  ovals  of,  338 
Catenary,  281 
Center  of  a  conic,  238 


Change  of  origin  without  change  of 
direction  of  axes,  217 

of  direction  of  axes  without  change 
of  origin,  221 

from  rectangular  to  oblique  axes 
without  change  of  origin,  224 

from  rectangular  to  polar  coordi- 
nates, 341 
Chord  of  contact,  248 
Circle,  134 

through  a  known  point,  136 

tangent  to  a  known  line,  136 

with  center  on  known  line,  137 

through  three  known  points,  138 

parametric  equation  of,  303 

involute  of,  311 

polar  equation  of,  342 

of  curvature,  354 
Cissoid,  151 
Classes  of  functions,  43 
Coefficient  of  an  element  of  a  determi- 
nant, 8 
Collinear  points,  38 
Complex  numbers,  31 

roots  of  an  equation,  82 
Components  of  velocity,  200 
Concavity  of  a  curve,  112 
Conchoid,  334 
Conic,  l48,  229 

classification,  237 

through  five  points,  241 

polar  equation,  343 
Conjugate  complex  numbers,  32 

axis  of  an  hyperbola,  145 

diameters,  258 

hyperbolas,  262 


381 


382 


INDEX 


Constant,  40 

of  integration,  206 
Contact,  point  of,  105 

chord  of,  248 
Continuity,  101 
Coordinate  axes,  35 

cliange  of,  217 
Coordinates 

rectangular,  35 

transformation  of,  217 

oblique,  223 

Cartesian,  224 

polar,  329 

relation  between  rectangular  and 
polar,  341 
Curvature,  353 

radius  of,  354,  360,  361 

circle  of,  354 

center  of,  356 
Curve,  Cartesian  equation  of,  44 

slope  of,  99 

degree  of,  166 

parametric  equations  of,  302 

polar  equation  of,  330 
Curves,  intersection  of,  161 
Curves  of  second  degree,  229 
Cycloid,  305 

Degree  of  a  curve,  166 
Depressed  equation,  79 
Derivative,  102 

of  a  polynomial,  97,  103 

sign  of,  106,  111 

second,  110 

higher.  111,  187 

theorems  on,  179 

of  w,  185 

illustrations  of,  203 

■with  respect  to  an  arc,  196,  347 
Descartes'  rule  of  signs,  87 

folium  of,  132 
Determinants,  1 

elements  of,  4 

minors  of,  4 

properties  of,  6 

expansion  of,  8 


Diameters,  of  a  conic,  252 

of  a  parabola,  254 

of  an  ellipse,  256 

of  an  hyperbola,  257 

conjugate,  258 
Differentiation,  102 

of  a  polynomial,  103 

of  algebraic  functions,  178 

successive,  187 

of  implicit  functions,  188 
Differentiation,  formulas  of 

for  a  polynomial,  103 

general,  184 

for  M»,  185 

for  trigonometric  functions,  272 

for  invei-se    trigonometric   func- 
tions, 276 

for  exponential  functions,  284 

for  logarithmic  functions,  284 

for  hyperbolic  functions,  290 

for  inverse  hyperbolic  functions, 
291 
Direction  of  a  curve,  197 

in  polar  coordinates,  345 
Directrix 

of  a  parabola,  146 

of  a  conic,  148 

of  an  ellipse,  149 

of  an  hyperbola,  149 
Discontinuity,  101 

examples  of,  41,  128,  268,  282 
Discriminant,  117 

of  a  quadratic  equation,  73,  117 

of  a  cubic  equation,  114,  117 

of  the  general  equation  of  the  sec- 
ond degree,  236 
Distance  between  two  points,  36 

of  a  point  from  a  straight  line,  63 

e,  the  number,  280 
Eccentric  angle  of  ellipse,  304 
Eccentricity  of  a  conic,  148 
Elasticity,  204 

Elements  of  a  determinant,  4 
Eliminants,  23 
Elimination,  1 


INDEX 


383 


Ellipse,  139 

referred  to  conjugate  diameters  as 
axes,  269 

parametric  representation  of,  303 
Energy,  kinetic,  203 
Epicycloid,  307 
Epitrochoid,  309 
Equation  in  one  variable 

solution  by  factoring,  77 

with  given  roots,  79 

depressed,  79 

number  of  roots  of,  80 

sum  and  product  of  roots  of,  82 

complex  roots  of,  82 

solution  of,  89 

Newton's  method  of  solution  of, 
114 

multiple  roots  of,  116 

resultant,  161 
Equation  of  a  curve,  44 
Equations  in  several  variables, 

linear,  1 

systems  of,  12 

homogeneous,  21 
Equations  in  two  variables 

of  first  degree,  52 

of  second  degree,  229 
Equations,  transcendental,  293 
Evolute,  357 
Expansion  of  a  determinant,  8 

coefficient  of,  204 
Explicit  algebraic  function,  188 
Exponential  functions,  279 

differentiation  of,  284 

Factoring,  solution  of  equations  by,  77 

of  quadratic  expressions,  79 
Factors  and  roots  of  an  equation,  78 

of  a  polynomial,  81,  83 
Foci  of  an  ellipse,  139 

of  an  hyperbola,  142 
Focus  of  a  parabola,  146 

of  a  conic,  148 
Folium  of  Descartes,  132 
Force,  202 
Function,  40 


Functional  notation,  44 
Functions,  classes  of,  43 

algebraic,  43,  121 

irrational,  44,  131 

transcendental,  44,  266 

defined  by  equations  of  the  second 
degree,  127 

involving  fractions,  128 

trigonometric,  266 

inverse  trigonometric,  269 

exponential,  279 

logarithmic,  279 

hyperbolic,  288 

inverse  hyperbolic,  291 

Graph,  40 

Graphical  representation,  28 

Harmonic  property  of  polars,  249 

division  of  a  line,  250 

motion,  275 
Homogeneous  equations,  21 
Horner's  method,  92 
Hyperbola,  142 

equilateral,  146 

referred  to  asymptotes  as  axes,  224 

referred  to  conjugate  diameters  as 
axes,  259 

conjugate,  262 
Hyperbolic  functions,  288 

inverse,  291 

differentiation  of,  290,  292 
Hyperbolic  spiral,  332 
Hypocycloid,  309 

four-cusped,  132 
Hypotrochoid,  309 

Imaginary  numbers 

(see  Number,  Complex) 
Implicit  algebraic  function,  188 
Increment,  100 
Infinity,  29,  128 
Inflection,  points  of,  112,  194 
Initial  line,  329 
Integration,  205 
Intercepts,  53 


384 


INDEX 


Interchange  of  axes,  223 
Intersection  of  curves,  161 

number»of  points  of,  169 
Involute,  357 

of  circle,  311 
Irrational  number,  28 

algebraic  functions,  44,  131 

roots  of  an  equation,  92,  114 
Isolated  point,  126 

Kinetic  energy,  203 

Latus  rectum,  211 

Limit  of  ratio  of  arc  to  chord,  195 

.  sin  ft       ,  1  —  cos  h    _^- 

of  and —,  270 

h  h 


of  (1  +  h)''  and 


1 


283 


Limiting  cases  of  a  conic,  234 
Limits,  97 

theorems  on,  178 
Locus,  45 

Locus  problems,  316 
Logarithm,  Napierian,  280 
Logarithmic  function,  279 

differentiation  of,  284 

spiral,  333 
Lemniscate,  340 
Lima^on,  336 

Maxima  and  minima,  108,  112,  192 
Minors  of  a  determinant,  4 
Momentum,  203 
Motion,  uniform,  199 

harmonic,  275 
Multiple  roots  of  an  equation,  116 

Napierian  logarithm,  280 

Newton  ■;S  method  of  solving  numerical 

equations,  114 
Normal,  64,  191 

Normal  equation  of  straight  line,  64  ' 
Number,  real,  28 
complex,  31 

Oblique  coordinates,  223 


Ordinate,  36 

Ovals  of  Cassini,  338 

Parabola,  146 

referred   to    tangent  at  ends  of 
latus  rectum,  132 

referred  to  a  diameter  and  a  tan- 
gent as  axes,  255 

cubical,  74 

semi-cubical,  131 
Parallel  lines,  56,  59 
Parametric  representation  of  curves, 

302 
Pedal  curves,  319 
Perpendicular  lines,  57,  59 
Plotting,  36,  329 
Point  of  division,  38 
Polar  of  a  point,  247 
Polar  coordinates,  329 
Polars,  reciprocal,  251 
Pole  of  a  straight  line,  247 

of  a  system  of  polar  coordinates, 
329 
Polynomial,  43 

of  first  degree,  50 

of  second  degree,  70 

of  nth  degree,  74 

factors  of,  81,  83 

derivative  of,  97 
\      square  root  of,  121 
Problems  on  straight  lines,  58 
Products,  graphs  of,  83 
Projection,  34 

Radical  axis,  175 
Radius  vector,  329 

of  curvature,  354,  360,  361 
Rate  of  change,  203 
Rational  algebraic  function,  43 

number,  28 

roots,  89 
Reciprocal  polars,  251 
Resultant,  23 

equation,  161 
Roots  of  an  equation 

and  factors,  78 


INDEX 


385 


Roots  of  an  equation 

number  of,  80 

sum  and  product  of,  82 

complex,  82 

location  of,  86 

rational,  89 

irrational,  92,  114 

multiple,  116 
Rose  of  three  leaves,  331 
Rotation  of  axes,  221 

Slope  of  a  straight  line,  64 

of  a  curve,  99 
Solution 

of  simultaneous  equations,  12 

of  algebraic  equations,  77,  89,  114 

of  transcendental  equations,  293 
Spiral 

of  Archimedes,  332 

hyperbolic,  332 

logarithmic,  333 
Straight  line,  50 

satisfying  two  conditions,  58 

normal  equation  of,  64 

parametric  equations  of,  302 

polar  equation  of,  342 
Strophoid,  152 
Subnormal,  210 

polar,  351 
Sub  tangent,  210 

polar,  351 
Supplemental  chords,  264 
Sylvester's  method  of  elimination,  24 


Systems  of  curves  with  common  points 
of  intersection,  171 

Tangent,  73,  84,  104,  190 

to  a  conic  with  given  slope,  163 

to  a  conic  at  a  given  point,  246 
Tractrix,  299 
Transcendental  functions,  44,  266 

equations,  293 
Transformation  of   coordinates,  217, 

341 
Transverse  axis  of  hyperbola,  145 
Trigonometric  functions,  266 

differentiation  of,  272 

inverse,  269 

differentiation  of,  276 
Trochoid,  306 
Turning  points  of  a  graph,  107 

Variable,  40 
Variation  of  sign,  87 
Vector,  radius,  329 
Vectorial  angle,  329 
Velocity,  198 

components  of,  200 
Vertex,  71 

of  a  parabola,  147 
Vertices 

of  an  ellipse,  141 

of  an  hyperbola,  144 

Witch,  149 
Zero,  29 


Date  Due 

m 

PRINTED 

IN  U.  S.  A. 

\    000  584  802    3 


The  RAIPH  D.  IKED  LIBRARf 


DEPARTMENT  OF  GEOLOGY 

UNIVKRSrrY  of  CALIFORNIA 

1-08  ANGKLES.  CALIF. 


